Resilver the mirror: OK, this is definitely the most involved option to resolve the problem. Resilvering is basically removing the protective, silver backing of the mirror and reapplying those layers again on the back of the glass. It is a more technical process (see a full tutorial here), and I think the larger the mirror, the harder it would be for a first timer. But it is an option if your mirror’s black spots are overwhelming and totally beyond any other repair (and actually, I think it looks like fun to try!).
Use aluminum foil: Say what?? Yep, this trick works really well if you have the kind of spots where the backing has been scraped off and you can see through to the other side of the glass. All you need to do is smooth out small pieces of aluminum foil and tape them with clear tape to the back of the mirror. The reflective foil is seen from the front and is much less noticeable when you hang it back up again. Since you probably already have tape and foil on hand, this is an easy solution for a few trouble spots.
Mirror-like spray paint: This is a similar option to the aluminum foil (this is what I used) and essentially does the same thing if you have the clear spot issue. If you have a bunch of small areas all over your mirror, it’s probably more efficient to just spray the whole back so any little spots get covered and you don’t have to foil 20 different areas.
Paint a frame to conceal trouble spots: A lot of times old mirrors will have the most spots around the very edge of the mirror because that’s where water got splashed or steam from a bathroom would creep behind the mirror over the years. While you can use the above tricks to fix or minimize damage on the edges, you can also tape off a border or design, use a few coats of spray paint, and create a new frame that covers up the spots instead! I’ve found that most spray paints will hold up just fine when cleaning the mirror with glass cleaner, but you can always buy a small craft mirror and test out your paint first if you want to make sure it won’t wipe right off during cleaning.