Streaks on a mirror are almost never dirt. They're leftover cleaner, or a dirty cloth spreading a thin film around. Use less product, use two clean microfiber cloths, and the streaks mostly disappear.

Spraying the cloth instead of the glass uses far less product, and it keeps liquid from running down behind the mirror's edge.
Mix equal parts water and white vinegar in a spray bottle. For a faster-drying version, use one part water, one part white vinegar and one part rubbing alcohol. Either one works on mirrors, windows, glass tabletops and picture-frame glass. Our guide to DIY cleaning solutions has other mixes.
Bathroom mirrors collect dried spots that glass cleaner just smears. Dab them first with a cotton pad dampened with rubbing alcohol. The alcohol dissolves hairspray, makeup and toothpaste residue in a few seconds. Then clean the whole mirror as above.
A mirror is glass with a reflective silver coating on the back. When liquid seeps behind the edge again and again, it eats into that coating and leaves black, spreading marks along the border. Those can't be repaired. Spraying the cloth instead of the mirror is the easiest way to prevent it, and wipe the bottom edge dry after you clean.
Wash them separately from towels, which shed lint onto them. Use regular detergent, no fabric softener and no dryer sheets, and air dry or tumble dry on low. A clean cloth is half the trick.
Usually too much cleaner, a dirty cloth, paper towel lint, or a cloth that was washed with fabric softener. Spray a little cleaner on a clean microfiber cloth, wipe, then buff dry immediately with a second dry cloth.
Equal parts water and white vinegar in a spray bottle. For faster drying, use one part each of water, white vinegar and rubbing alcohol.
Dab the spots with a cotton pad dampened with rubbing alcohol before cleaning the whole mirror. The alcohol dissolves the residue that glass cleaner just smears.
Liquid seeping behind the edge damages the silver coating on the back of the mirror. It can't be repaired, but spraying the cloth instead of the mirror prevents it.