Have you tried window cleaner? If the silver in the back is failing that is another issue entirely and it will need to be resilvered or replaced, as the case may be.
I have an antique mirror that has a few discolored spots, but the main thing is it looks like someone tried to clean it with a dirty cloth. I know it needs to be resilvered, but I was told a long time ago that an antique is devalued if resilvered, so I just enjoy it the way it is. If your furniture is collectible, place another mirrow somewhere that you can use. If not, take it off and hang a suitable mirrow above the dresser.
Rubbing alcohol. It's AMAZING for cleaning filmy mirrors!
If it is a old mirror the silver might need to be re-silvered. No amount of cleaning will work. If it is a fairly new mirror then it could be the products you have used on it before that have left a film. Including window cleaners. Alcohol and vinegar do work great on this. A glass cloth which is a smooth lint free cloth. Spray the cleaner onto the cloth never on the mirror. Clean and dry. While this works great on cleaning build up I wouldn't use it all the time. Because most people spray cleaners on mirrors directly and they get into the corners of frames and it etches or the mirrors because you can't get into the tiny owners of the frame and you can't dry it.
If this is a old mirror re-silvering costs a lot so I would figure if it was valuable enough to warrent spending that kind of money.
I finally ended up using nail polish remover and it worked!
If you ever have film on your mirror you can use rubbing alcohol to get it off. It's a great window and mirror cleaner!