The hinge can't be fixed and it looks like it may have been riveted to the mounting board before the mirror was glued on. Ordinarily in this situation you would have a new mirror cut and mounted on a new board, and install new hinges. But that doesn't look like any hinge I've ever seen; it looks like a specialty design specific to your medicine cabinet. I have a similarly mounted vanity mirror, but it's attached to 3/4" plywood, and this whole thing is mounted with Euro hinges to a wood-framed cabinet, which is probably not a possibility in your situation. If you can post another picture or two to show what your cabinet looks like, I may be able to offer some more helpful advice. But at this point I'm not optimistic there is an easy fix.