What would we see if we hold the mirror in front of us and behind us.
If you are traveling at near lightspeed, and the mirrors are traveling with you, there will be no change in the behavior of the mirrors, or any other observation you make of anything else moving at near lightspeed with you. Relativity tells that light must travel at the same speed in any reference frame, and a reference frame traveling at nearly the speed of light is no different.
When you are traveling almost at the lightspeed, you are moving at this speed only as compared to another frame of reference (body). Which means that - according to the postulates of the Theory of Special Relativity - you are allowed to assume that you are not moving at all and that this other body is moving at near-lightspeed (but in the opposite direction). If the movements are inertial - as SR assumes - then there is no way to tell who is moving: you or the other body.
The conclusion is that according to SR there can be no difference whatsoever regarding your image in the mirrors - whether you are moving or not. If there were, you could claim that there some absolute state of rest can be found, which SR denies to exist.
So if you emit light toward any of the mirrors, and measure its speed upon the return of the beam, you will always measure $c$.
The speed of light is completely dependent on the frame of reference you are using. As already stated, the mirrors traveling with you are in your frame of reference and, therefore, will not be affected by the tremendous speed that you are traveling.
It may help to think of the frame of reference that you are in right now, regarding your position on Earth traveling around the sun, which is rotating around our galaxy, which is traveling through space at a tremendous speed itself. All of this motion does not affect the speed of light being emitted from your computer screen, for example, since your computer is in your frame of reference.