To answer this we need to set up an inertial frame of reference from which to see what happens. An inertial frame of reference is just equivalent to measuring things relative to a freely falling platform. Let's say our inertial frame of reference is our friend Bob who is in orbit (free fall) about Earth at some distance and not moving particulaly fast with respect to things in the Solar system. He looks out of the window of his spaceship and detects me go by at the speed of light busily looking into my mirror, closely followed, but not being caught, by a beam of light. Bob uses Special Relativity to show that to him my clock appears to have stopped - I am frozen into immobility. Light leaving my face is moving at the speed of light (measured by Bob) and that light travelling directly forwards is just ahead of me, frozen in space (relative to me) trying but never managing to reach that mirror which is moving away at the same speed. Now, assume I turned off the drive - I don't change speed immediately (see Newton's First Law) - I'm still going at the same speed (The speed of light according to Bob's recent measurement). I'm an inertial frame of reference now. I noticed Bob just now. His clock seems to have stopped and he's just frozen there. He looked as though he was about to make a speed measurement. He was going like a bat out of hell at the speed of light. I look behind me for that beam of light and I see it's the light of a torch that someone is shining. It's shining in my eyes ... it's coming at me at a speed of ... let me see, yes, the speed of light. Oh, someone just turned it off. I look back at the mirror ... nothing special - still as handsome as ever! ... Part of the puzzle is that you can't talk about "speed" unless you measure it between two moving objects. I need Bob in order just to define my speed in the first place. Part of the puzzle is that time is modified. I get through an infinite number of Bob's seconds as I go past, so he doesn't see the beam of light approach me as I do, and vice versa.

A monkey? Just kidding. You would see yourself. Light travels at lightspeed no matter from where. The rays would hit the mirror at lightspeed and return at lightspeed. Nothing adds to the speed of light.