How can the momentum-position Uncertainty Principle not violate conservation of energy? As far as I understand, the primary reason why an electron doesn't fall into a nucleus is that, when it gets close, it has a sufficiently high probability of having a lot of momentum to get back to its original position. This high probability, in my understanding, arises because the electron is confined to only a small amount of space around the nucleus, and hence, by Heisenberg's Uncertainty Principle,the uncertainty in its momentum is high, which in turn increases the probability of the momentum being large.
In such cases, where the electron "bounces off" or "phases through" the nucleus, the momentum would always have a non-zero magnitude, and hence the kinetic energy would always be non-zero. This way, energy would be gained with each new "bounce". Additionally, the electrons would be constantly accelerating, radiating away light of all frequencies.
Is my premise incorrect, or is it the implication? And in what way is it incorrect?
 A: Don't take this the wrong way, but it's pretty much incorrect start to finish because it applies classical reasoning to quantum mechanical systems.  Sometimes semi-classical reasoning can help give us a bit of intuition, but this is not one of those times.
The real answer to this question is simply that electrons are not little balls orbiting a nucleus like planets around the sun.  Electrons bound to an atom do not have well-defined momenta, nor do they have well-defined positions, and the spreads in the possible results of position and momentum measurements obey the uncertainty principle.
Additionally, the conservation of energy takes on a somewhat different character in quantum mechanics because quantum mechanical particles do not generically have well-defined energies either.  It's true that for certain systems (i.e. those which can be described by Hamiltonian operators with no explicit time-dependence) the expectation value of the energy is conserved as the state evolves, but this doesn't correspond to the actual energy of the system at any point in time.
