unfortunately, this is a time consuming process. mirrors are made by floating the hot glass on a liguid metal bath... almost any scratch will show.

there is a type of glass polish called a "spit shine" meaning you polish over the scratches, but not remove the scratches.

there is an optical polish, which is removing all the scratches... 800 grit 1000 grit, 1500 grit, 1800 grit, heavy compound, medium compound, light compound.


tips for trying to lessen and remove scratches:
mask off the rest of the mirror, dust from this process will create more scratches.
try using some medium or fine buffing compound first, this may hide or at least take some whiteness out of the scratch.
work with a little bit of sunlight coming into a room-seeing the scratch is sometimes hard.
sell the mirror fast with a discount-the eye can catch almost any flaw