shim it, thats what i would do, unless a replacement was easy to perform and cost under $10.

Get some small piece silicone putty (silly putty) and jam it in there (there being around the mirror face, around the edges).

Or wrap something stiff and water proof like playing cards in electrical tape and jam it in a corner.

Or if you've got a thin silicone rubber band, stretch it around the mirror or jam thin sheet underneath (natural rubber will melt).

What might also work is flexible caulking (bathroom silicone),
but application would be trickier and it might stick too much.