Where does rest energy fit into the calculation of total energy? Electron energy is often expressed as the sum of potential energy and kinetic energy : $$E = U + KE$$ 
In this case, where does the rest energy $511\ \text{keV}$ fit in? 
How can I reconcile these two equations? 
$$\text{Total energy} = \text{Kinetic Energy} + \text{Potential Energy}$$
$$\text{Total energy} = \text{Kinetic Energy} + \text{rest energy}$$
 A: Actually rest energy (or $mc^2$) is a third form of energy.
So you can reconcile the equations like this:
$$\text{Total energy} = \text{Rest Energy} + \text{Kinetic Energy} + \text{Potential Energy}$$
In mechanics, chemistry and atomic physics the masses don't change.
Hence the rest energy, although being very large, doesn't
need to be considered.
The situation becomes different for other processes
(radioactive decay, nuclear fission, nuclear fusion,
pair production, electron-positron annihilation),
where particles get converted to other particles.
Here the rest energy needs to be considered to explain energy conservation.
A: 
Electron energy is often expressed as the sum of potential energy and
  kinetic energy : E=U+KE

It is expressed that way in non relativistic QM. The Hamiltonian of the Schroedinger equation is exacly that, where $KE = \frac{1}{2}mv^2 = \frac{p^2}{2m} = -\hbar \frac{\nabla^2}{2m}$.
Dirac managed to find a corrected relativistic equation from the expression $E^2 = p^2c^2 + m^2c^4$. If the momentum is zero, it is only $E = mc^2$. So, it takes in consideration the rest energy or the mass.
It was one of the best confirmations of the special relativity, because some properties that were known only from experience as spin came up from the maths. 
