How do we know the LHC results are robust? Nature article on reproducibility in science.
According to that article, a (surprisingly) large number of experiments aren't reproducible, or at least there have been failed attempted reproductions. In one of the figures, it's said that 70% of scientists in physics & engineering have failed to reproduce someone else's results, and 50% have failed to reproduce their own.
Clearly, if something cannot be reproduced, its veracity is called into question. Also clearly, because there's only one particle accelerator with the power of the LHC in the world, we aren't able to independently reproduce LHC results. In fact, because 50% of physics & engineering experiments aren't reproducible by the original scientists, one might expect there's a 50% chance that if the people who originally built the LHC built another LHC, they will not reach the same results. How, then, do we know that the LHC results (such as the discovery of the Higgs boson) are robust? Or do we not know the LHC results are robust, and are effectively proceeding on faith that they are?
EDIT: As pointed out by Chris Hayes in the comments, I misinterpreted the Nature article. It says that 50% of physical scientists have failed to reproduce their own results, which is not the same statement as 50% of physics experiments aren't reproducible. This significantly eases the concern I had when I wrote the question. I'm leaving the question here however, because the core idea - how can we know the LHC's results are robust when we only have one LHC? - remains the same, and because innisfree wrote an excellent answer.
 A: The paper seems to be a statistical analysis of opinions, and in no way is rigorous enough to raise a question about the LHC. It is statistics about undisclosed statistics.
Here is a simpler example for statistics of failures: Take an Olympics athlete.  How many failures before breaking the record? Is the record not broken because there may have been a thousand failures before breaking it?
What about the hundreds of athletes who try to reproduce and get a better record? Should they not try?
The statistics of failed experiments is similar: There is a goal (actually thousands of goals depending on the physics discipline), and a number of trials to reach the goal, though the  olympics record analogy should not be taken too far, only to point out the difficulty of combining statistics from a large number of sets. In physics there may be wrong assumptions, blind alleys, logical errors... that may contribute to the failure of reproducibility. The confidence level from statistical and systematic errors are used to define the robustness of a measurement. 
from the question:
"because 50% of physics & engineering experiments aren't reproducible by the original scientists",
This is a fake statement from a dubious poll. The statistical significance of the "not reproducible " has not been checked in the poll. Only if it were a one standard deviation result , there exists  almost a 50% a probability of the next trial not to reproduce. 

one might expect there's a 50% chance that if the people who originally built the LHC built another LHC, they will not reach the same results

No way, because engineering and physics analysis at the LHC are over the 4 sigma level, and the probability of negation is small. Even a 3sigma level has confidence 99% , so the chance is in no way 50%.
We know the LHC results are robust because there are two major and many smaller experiments trying for the same goals. The reason there are two experiments is so that systematic errors in one will not give spurious results. We trust that the measurement statistics that give the end results are correct, as we trust for the record breaking run that the measured times and distances are correct.
(And LHC is not an experiment. It is where experiments can be carried out depending on the efforts and ingenuity of researchers, it is the field where the Olympics takes place.)
The robustness of scientific results depends on the specific experimental measurements,  not on integrating over all disparate  experiments ever made. Bad use of statistics. For statistics of statistics, i.e. the confidence level of the "failed experiments" have to be done rigorously and the paper is not doing that.
Another way to look at it: If there were no failures , would the experiments mean anything? They would be predictable by pen and paper.
A: Any one experiment is repeated many times on the same equipment. They look for rare events, and it takes a lot of rare events to be sure that they aren't just coincidence.
The question about how many LHCs it takes to be sure, is different.
Each LHC component had to be carefully tested to make sure it was in spec. Remember the example of the experiment that seemed to get a result slightly faster than light. Because it was so important, they went to great expense to test everything, components around the world, until they found two components that were out of spec, that created the small error. If the error had been in the other direction would they have done that testing? No. They wouldn't even notice the error. It wouldn't be important. What made this one important was faster than light. Did they carefully record every out-of-spec component they found that would tend to slow the signal, that might cancel the positive errors they found? Maybe. That wasn't what they were looking for, though. That was a complication and not a solution to the problem.
After the tested LHC components are installed they must be tested again in case they were changed while being handled. 
Then they must be calibrated. Every analog output could have a baseline that's a little bit off, because of random things. A solder joint that's slightly different. An AC circuit nearby that changes things a little bit every 120th of a second. The baseline must be calibrated for every one of them. Once the signal has been converted to digital then it's OK. Errors smaller than the cutoff are ignored, and larger errors make one bit difference. For the calibration, you know what the outcome is supposed to be, so you set it to that. 
Could all this have somehow changed the outcomes so that some extremely unlikely results are falsely reported more often than they should be?
There's no theoretical reason to expect it. And the engineers who assembled the LHC were very very careful. But how could we test it? The obvious way is to build at least 2 more LHCs and notice how consistent their results are. That would be very expensive. It will not be done.
We can get some confidence by looking at results from other machinery. It's like -- the LHC was used to scan for a wide range of possible results that could be called the Higgs boson. They could do in years what a lesser machine might take centuries to do. But once we have a specific Higgs boson to look for, some of the others can look for that specifically and see whether they find it. If they do, then there's probably something there beyond equipment error.
Something else they can do (which I think they are doing part of the time) is look for things that are supposed to not happen that nobody predicts will happen. When they find one for sure then everybody will get excited. People will say there's something wrong, and insist that they check for every possible error that could give them that result. Like with the faster-than-light thing. 
A: Not only are the LHC experiments reproduced many times over, but iirc, for important results they actually have two independent teams working on different versions of the same experiment. 
It worth noting however that the experimental physicists often take a different, and more rigorous, approach to their results than theorists. For example, theorists have been keen to claim that the predicted quark-gluon plasma had been produced, but experimental groups are more cautious about jumping to conclusions, saying, for example, 

“However, detailed analyses of the data also make it clear that this hot, dense medium has properties that are surprising, and not yet fully understood in terms of the early expectations for the quark gluon plasma.” --- Relativistic Heavy Ion Collider, Brookhaven National Laboratory, 2005, Hunting the Quark Gluon Plasma, results from the first three years. Formal Report, BNL -73847-2005

It appears that the new form of matter behaves more like a liquid than a plasma, as though composed of constituent particles which tend to flow together rather than randomly,
as predicted by asymptotic freedom. In the absence of viable calculations showing
exactly how quark soup should behave, it is difficult to assess the
implications for quantum chromodynamics.
Likewise, while it is clear that they have found a particle called the Higgs, there is precisely no empirical evidence to relate that particle to the unobservable Higgs mechanism created by theorists.
A: In addition to innisfree's excellent list, there's another fundamental difference between modern physics experiments and human-based experiments: While the latter tend to be exploratory, physics experiments these days are primarily confirmatory.
In particular, we have theories (sometimes competing theories) that model our idea of how physics works. These theories make specific predictions about the kinds of results we ought to see, and physics experiments are generally then built to discriminate between the various predictions, which are typically either of the form "this effect happens or doesn't" (jet quenching, dispersion in the speed of light due to quantized space), or "this variable has some value" (the mass of the Higgs boson). We use computer simulations to produce pictures of what the results would look like in the different cases and then match the experimental data with those models; nearly always, what we get matches one or the other of the suspected cases. In this way, experimental results in physics are rarely shocking.
Occasionally, however, what we see is something really unexpected, such as the time OPERA seemed to have observed faster-than-light motion—or, for that matter, Rutherford's gold-foil experiment. In these cases, priority tends to go toward reproducing the effect if possible and explaining what's going on (which usually tends to be an error of some sort, such as the miswired cable in OPERA, but does sometimes reveal something totally new, which then tends to become the subject of intense research itself until the new effect is understood well enough to start making models of it again).
A: That's a really great question. The 'replication crisis' is that many effects in social sciences (and, although to a lesser extent, other scientific fields) couldn't be reproduced. There are many factors leading to this phenomenon, including


*

*Weak standards of evidence, e.g., $2\sigma$ evidence required to demonstrate an effect

*Researchers (subconsciously or otherwise) conducting bad scientific practice by selectively reporting and publishing significant results. E.g. considering many different effects until they find a significant effect or collecting data until they find a significant effect.

*Poor training in statistical methods.


I'm not entirely sure about the exact efforts that the LHC experiments are making to ensure that they don't suffer the same problems. But let me say some things that should at least put your mind at ease:


*

*Particle physics typically requires a high-standard of evidence for discoveries ($5\sigma$). To put that into perspective, the corresponding type-1 error rates are $0.05$ for $2\sigma$ and about $3\times10^{-7}$ for $5\sigma$

*The results from the LHC are already replicated! 


*

*There are several detectors placed around the LHC ring. Two of them, called ATLAS and CMS, are general purpose detectors for Standard Model and Beyond the Standard Model physics. Both of them found compelling evidence for the Higgs boson. They are in principle completely independent (though in practice staff switch experiments, experimentalists from each experiment presumably talk and socialize with each other etc, so possibly a very small dependence in analysis choices etc). 

*The Tevatron, a similar collider experiment in the USA operating at lower-energies, found direct evidence for the Higgs boson.

*The Higgs boson was observed in several datasets collected at the LHC


*The LHC (typically) publishes findings regardless of their statistical significance, i.e., significant results are not selectively reported.

*The LHC teams are guided by statistical committees, hopefully ensuring good practice

*The LHC is in principle committed to open data, which means a lot of the data should at some point become public. This is one recommendation for helping the crisis in social sciences.

*Typical training for experimentalists at the LHC includes basic statistics (although in my experience LHC experimentalits are still subject to the same traps and misinterpretations as everyone else). 

*All members (thousands) of the experimental teams are authors on the papers. The incentive for bad practices such as $p$-hacking is presumably slightly lowered, as you cannot 'discover' a new effect and publish it only under your own name, and have improved job/grant prospects. This incentive might be a factor in the replication crisis in social sciences.

*All papers are subject to internal review (which I understand to be quite rigorous) as well as external review by a journal

*LHC analyses are often (I'm not sure who plans or decides this) blinded. This means that the experimentalists cannot tweak the analyses depending on the result. They are 'blind' to the result, make their choices, then unblind it only at the end. This should help prevent $p$-hacking

*LHC analysis typically (though not always) report a global $p$-value, which has beeen corrected for multiple comparisons (the look-elsewhere effect).

*The Higgs boson (or similar new physics) was theoretically required due to a 'no-lose' theorem about the breakdown of models without a Higgs at LHC energies, so we can be even more confident that it is a genuine effect. 
The other new effects that are being searched for at the LHC, however,  arguably aren't as well motivated, so this doesn't apply to them. E.g., there was no a priori motivation for a 750 GeV resonanace that was hinted at in data but ultimately disappeared.


If anything, there is a suspicion that the practices at the LHC might even result in the opposite of the 'replication crisis;' analyses that find effects that are somewhat significant might be examined and tweaked until they decrease. In this paper it was argued this was the case for SUSY searches in run-1.
