When cleaning a bathroom mirror, dampen a large sponge with hot water. Wipe your mirror down completely, beginning at the top and working your way to the bottom until the entire mirror is moist. This will help the mirror cleaning product (your active agent) fight the dirty marks.
With an ammonia-based cleaning product, spray the entire mirror. Wipe down the mirror with the sponge (you may want to wring it out first so that it is not sopping wet). Again, begin at the top and work your way down, making sure all of the mirror is covered with cleaning agent.
Remove Excess Water
Wipe excess water with a squeegee (one that you may use to remove frost on your car or fog on your shower). Move the squeegee from top to bottom, making sure your mirror is dry.
Wipe It Down
Using a linen cloth (paper towels and bath tissue can leave behind lint), wipe the entire mirror. Begin at the top and work your way down. Alternatively, you can use newspaper.
While this may be a good way to clean your bathroom mirror, you might also consider making your own cleaning agent by rubbing your mirrors down with vinegar instead of an ammonia-based product. You can also try a solution of 2 cups rubbing alcohol, 2 cups water and 2 tablespoons dishwashing soap. Put it in a spray bottle and use as your cleansing agent.