So an electron can only orbit a nucleus where its wavelength makes a standing wave, leading to discrete energy levels in atoms.
That is the Bohr model which has been superseded by quantum mechanics. In quantum mechanics the electron occupies definite energy levels which arise because of the potential well that is generated by the nucleus. As the other answers say, it has not an orbit with a momentum, but an orbital, a locus in space that gives the probability density of finding it if trying to measure it.
But the de Broglie wavelength of an electron = h/mv.
This works for an electron as a free particle, not bound in a potential well where its probabilistic-wave nature dominates.
So the wavelength of an electron changes with velocity of an electron.
Of a free electron.
So are there different energy levels for electrons of different velocities (and wavelengths?) Or do all electrons in atoms have the same velocity? (and if so why?)
The electrons around the nucleus of an atom are in different energy levels that can be calculated by using the potential of the potential well. Due to the Pauli exclusion principle no two electrons can have the exact energy level .
To answer the title:
Can an atoms energy levels be changed by changing the de Broglie wavelength of electrons?
The energy levels are fixed by the potential and always have a width given by the wavefunction at that level.