Why does the Heisenberg uncertainty principle work in an atom? Where is my logic wrong?

An electron can only assume quantized energies in an atom.

If an electron is localized in space in a hydrogen atom, its radius is known. If its radius is known, its electrostatic potential energy is known. Then, if the electron assumes quantized energies, its kinetic energies are quantized by $E_\mathrm{total} - E_\mathrm{potential}$ for all quantized possible total energies of the electron. This gives us quantized kinetic energy and thus momentum, giving us quantized frequencies. If you sum all of the waves of those frequencies, will it give the localized position wave-function? They can't right due to the fact that they're quantized.

This is probably very wrong; I just don't understand how.
Electrons don't have a well-defined orbital radius even in a hydrogen atom

An electron "orbiting" in a hydrogen atom does not have a well-defined position or even just a well-defined radius just because it has a well-defined energy.

The orbital is more like a 3D probability density function covering a cloudy space of a defined shape (spherical in the case of the lowest energy orbital but not of a single radius and extending with a finite probability from the nucleus to a vaguely defined boundary we often regard as the radius of the atom).

Different orbitals have different, increasingly complex, shapes defined by solutions of the Schrodinger equation. Heisenberg's uncertainly relationship limits our ability to define the location and momentum of the electron at the same time. The fuzzy probability clouds defined by Schrodinger's solutions are the best we can do.