The Bohr problem has just radius, this problem has two radii.
At the moment you decide the system is quantum mechanical, you cannot have a classical trajectory anymore. You cannot see or define a single revolution path anymore, rather you can describe it statistically - e.g. how much motion system has in separate axes. Every measurement trying to reconstruct the ellipse is limited by Heisenberg uncertainty, in that sense there are two degrees of freedom.
- you strictly stick to the ellipse (you use just one degree of freedom) it will surely not reflect the quantum mechanical motion, but represent some artificial input condition - and you will go the Bohr direction with just one quantum number and probably some like-deformation parameter described by excentricity $\epsilon$. Or you should find a way to quantize $\epsilon$.
Did you check
http://uw.physics.wisc.edu/~knutson/phy448/wilson-sommerfeld.pdf ? They speak about the rule that the action integral of a generalized coordinate and its conjugate momentum taken over one cycle of motion is quantized (if cyclic motion).
Equation of motion is from 4.6 - 4.8. where they get the equations you show.