In your case, $\bar x_{JK}$ and $\sigma_{JK}^2$ should be exactly equal to $\bar x$ and $\sigma$. You can show that by taking equations (4) and (5) and plugging in the Jackknife definitions (and the definition of $\bar x$) and doing some (perhaps ugly) algebra. Of course $\sigma_{JK}^2$ should become smaller with increasing $N$, but so should $\sigma^2$.
At first and second glance I couldn't spot any error in your equations, so maybe you're right and there's a bug in the implementation.
In general the Jackknife results (with or without bias correction) will be identical to the results of the usual formulas for mean and standard error as long as you're computing the plain average of some function $f$,
$$ \bar f = \frac{1}{N} \sum_{i=1}^N f(x_i). $$
(Note that $x_i$ can be a vector or something else, too.)
In our collaboration we use to call these “primary observables”, but I don't know if that's standard terminology. So in these cases there's no point in going through the Jackknife procedure.
This holds for any function $f$, be it $x^5$, $\sin(x)$, $x/|x|$ or whatever. Also for linear combinations of primary observables the Jackknife results agree, since you can pull the linear combination into the average, e.g.
$$ 2 \bar f - 3 \bar g = \overline{2 f - 3 g}. $$
So in effect it's a primary observable itself.
Where Jackknife does make a difference is if you have a non-linear function of primary observables. (We call these “derived observables”.) An example is the Binder cumulant (see OP's comment), the ratio of the expectation value of $x^4$ and the square of the expectation value of $x^2$,
$$ \frac{\langle x^4 \rangle}{\langle x^2 \rangle^2}. $$
(I'm using the angle brackets here to denote the actual mathematical expectation values in contrast to the average of a finite sample.)
A natural estimator would be
$$ \hat\theta = \frac{\overline{x^4}}{\left( \overline{x^2} \right)^2} = \frac{\frac{1}{N} \sum_{i=1}^N x_i^4}{\left( \frac{1}{N} \sum_{i=1}^N x_i^2 \right)^2}. $$
$\overline{x^4}$ and $\overline{x^2}$ considered separately are primary observables, so you can estimate their standard errors in the usual way without Jackknife. To estimate the standard error of $\hat\theta$ you could then use simple error propagation,
$$ \sigma_{\hat\theta}^2 = \left( \frac{\partial\hat\theta}{\partial\overline{x^4}} \right)^2 \cdot \sigma_{\overline{x^4}}^2 + \left( \frac{\partial\hat\theta}{\partial\overline{x^2}} \right)^2 \cdot \sigma_{\overline{x^2}}^2, $$
but this approach has several problems:
The estimator $\hat\theta$ may be biased, i.e. for finite sample sizes $N$ it may yield results that are too large or too small on average. This is mostly overcome by the Jackknife method as it removes the part of the bias that goes with $1/N$. However, according to OP's comment and my own experimentation this doesn't seem to be a big issue here anyway and one can do without the bias correction.
The error estimator $\sigma_{\hat\theta}$ effectively linearizes $\hat\theta$, because it uses only its first derivatives. It ignores contributions from higher-order derivatives. Sadly, Jackknife doesn't help here, either, but again the effect is small anyway.
The most significant problem is that the error estimator ignores the correlation between $\overline{x^4}$ and $\overline{x^2}$. If you happen to have a sample where the average $x^2$ is larger than usual, the average of $x^4$ is probably larger, too. In the ratio, both deviations would cancel partially. So the actual error of $\hat\theta$ will be smaller than the result of simple error propagation. Jackknife automatically takes care of these correlations and should give a much more accurate estimate of the standard error.
Another potential problem is auto-correlation, i.e. correlation of the $x_i$ at different $i$, as it would arise in a typical Monte Carlo simulation based on Markov chains. This will make the error estimate smaller than the actual error. In contrast to the correlation of $\overline{x^4}$ and $\overline{x^2}$, Jackknife doesn't take auto-correlation into account at all. Instead one often uses binning to deal with auto-correlation, i.e. one divides the $x_i$ into consecutive groups of equal size, takes the average of each group and analyzes the averages instead of the original $x_i$. Alternatively one can use a variant of Jackknife where consecutive groups of $x_i$ are removed from the sample instead of a single element at a time, but I couldn't find a good reference for that.