Why doesn't the energy of an incident photon impact the angle of reflection? Note that I am not asking why the angle of reflection is what it is. That is a different question.
I am asking why the energy of the incident photon does not affect the interaction in question at all, which is not the case for light interacting in other cases, such as Compton Scattering, and when light traverses certain mediums.
It's actually quite strange when you think about it, because even if you view the interaction as the result of a "collision" of sorts between the incident light and the mirror, you'd think, intuitively, that a higher energy light source would have a more pronounced interaction with the surface of the mirror, producing some kind of different behavior, but that's clearly not what happens.
In short, why does the equation of interaction that governs reflection require less information than the equation of interaction that governs light traveling through a prism? Or Compton Scattering?
In all cases, what must be taking place is an exchange of momentum between the photons and some other system, yet in the case of reflection, the total momentum of the photon is irrelevant, whereas in Compton Scattering, the total momentum of the photon is relevant, yet the same physical mechanics must be taking place in each case.
Moreover, in Compton Scattering, the photon loses energy. In reflection, it cannot be the case that any meaningful energy is lost by the photon, otherwise mirrors wouldn't reproduce images accurately.
What is driving this difference in behavior? 
It seems that light has three ways of interacting:
Exchanging momentum in a manner in which its total momentum is irrelevant (reflection);
Exchanging momentum in a manner in which its total momentum is relevant, but no energy is lost (refraction); and
Exchanging momentum in a manner in which its total momentum is relevant, and some energy is lost (Compton Scattering).
What I'm looking for is an explanation as to why a photon would require more than one equation of interaction for when it exchanges momentum with another system.
 A: Energy (e) can basically be defined as:
$e=$h$\nu$
So, energy depends on frequency. Now, think, will the frequency of a given light ray change on reflection? It doesn't do so even in refraction across two different media. Since, the frequency of light doesn't change, its colour won't change, and so, its energy will also not change.
Also, since colour depends on energy of   the photons of the light ray, you won't see the change in colour after reflection or even after refraction.
Coming back to your question, In Compton scattering, the electron's momentum change involves a relativistic change in the energy of the electron, so it is not simply related to the change in energy occurring in classical physics. The change of the magnitude of the momentum of the photon is not just related to the change of its energy; it also involves a change in direction. The equations of Compton scattering are also true for photons of you look at it in a quantum mechanical way. It interacts with the electrons present in the surrounding medium. The effect of decrease in energy of the photon in this case will be negligible, due to negligible scattering by the electrons of the surrounding medium. So, in a classical analysis, we can neglect such effects. Also, one difference could be that the electrons in a mirror are confined to their conduction band. The momentum is transferred without transfer of energy because the energy derived from the photons is insufficient to eject out electrons from the conduction band. This means that more than a single electron is involved in the interaction.
A: You have asked several questions here. 
(1)The title question asks about the angle of reflection.
(2) Then you ask why doesn’t the energy of a photon affect the interaction of a reflection?
(3) why do photons in Compton scattering loose energy but not during reflection?
(1) The angle of reflection is not determined by single photons. Billions of photons are absorbed across the surface and  re- emitted in random directions.
(2) The energy of individual photons DOES affect the colors of the reflection.
(3) During reflection photons are absorbed raising the energy levels of electrons. Then new photons are emitted with a frequencies based on those energy levels. During Compton scattering the photons are not absorbed. Instead they barely skip off the edge the electrons and loose some of their energy as their frequencies are slowed down.
