Adding to what DaleSpam said: If the mirror has zero velocity relative to you, you get the same result no matter what your speed is relative to something else (like the house you live in). We don't have to worry about the speed of the "light source" too much, because in this case you are the light source. The light you see in the mirror is light that's been scattered off your clothes and the parts of your body that aren't covered with clothes. If the only light that hits you comes from a lamp that has a very high velocity relative to you, the wavelength of the light might be shifted out of the range that your eyes can detect, but in this case, you won't see your hand in front of you either.
It isn't possible to completely explain why you can see yourself. Questions about how the world works can only be answered by theories*, and every theory is based on a set of assumptions that defines the theory. In this case, the fact that the speed of the light that scatters off you will have speed c relative to you, regardless of your speed relative to anything else, is something that follows almost immediately from those assumptions. There's also some freedom in how those assumptions are chosen, and if we want to, we can have this fact be one of the assumptions. The ultimate justification for it is that the theory makes predictions that are more accurate than the predictions of the theories in which space and time have properties that agree with your intuition.*) Contrary to popular belief, a theory isn't "a guess that might be true". It's a set of statements that can be used to make predictions about results of experiments. If the predictions are accurate, it's a good theory. If they're not, it's a bad theory. If it doesn't make predictions (like the claim "God exists"), it's not a theory.
