How do I fix an antique dresser mirror that is looking fuzzy?

You could try some of the new mirror paint from Krylone or Rustoleum, but it will still have an antiqued appearance. You could also look for a company that does mirror re-silvering restoration.... call some glass companies to find one.

https://www.hunker.com/12002132/how-to-repair-a-scratch-in-reflective-coating-on-a-mirror

https://rearviewmirrorglue.com/how-to-repair-mirror-desilvering/

It sounds like it has desilvered or in other words is losing the coating on the back of the mirror. You can replace the mirror, have it professionally resilvered, or treat it as a sign of the mirror's age.

My Mom taught me this & have done it several times: remove the mirror from frame; the silver can be removed with a cleaner such as Lime-Away or citrus paint stripper; then have a new mirror (very thin glass) cut to same dimensions.

Clean both well & replace the antique clear former mirror with the new mirror behind it back into the frame.

Here you go with a good video.. https://www.youtube.com/watch?v=kBQ1RhKd3B0

Have it resilvered.

Clean the mirror with glass cleanser, shaving cream, or vinegar. Once you've removed dirt and gunk from the mirror, you'll be able to see better where the mirror is clear or cloudy. Then wipe the surface of the mirror with glass cleanser, shaving cream, or vinegar until the cloudiness disappears. If there is still a problem will will most likely need to have it silvered again by a professional.