How do you remove double-sided tape?

Try to peel it off first, and what's left I suggest soaking it in a solution like water & alcohol, and to get the glue off try scrubbing with a soft steel wool pad (such as those to clean algae infested fish tanks).

If the tape glue is really stuck (or you have an expensive mirror you don't want ANYTHING to happen to) there is a product called Goo-Gone which I LOVE. It gets stuff like tape, crayon (if you have kids or were acting like a kid) and all that other stuff off of pretty much anything. It's kinda oily and you have to make sure all the residue is off of your mirror to prevent streaks - but it's really worth it. It gets bumper stickers off Windows in trucks and it works perfectly. See related link for more on this product