For hanging a proper wash kit up off a mirror (or glossy tile), I use the kind of vacuum forming suction cup used for removing iMac and similar screens (or pulling out dents in a car). They are super strong, small enough to drop in the wash kit, and will NOT fall off (just remember to release the vacuum otherwise you're taking the mirror off with you... they are very strong!).
(search for iPhone iMac suction cup and you should find small ones like this)
