The way I do it is to place the mirror in a perferated vegie bowl, at an angle
of about 45 degrees to the bench top. I then place this in the sink and start
the cold tap flowing so the water has a steady flow over the entire mirror
surface. You may have to adjust the mirror angle and flow of water to get
this right. Although it shoudln't be blasting the mirror (if it starts to splash off
the surface, your using to much power) a steady flow is required. I leave this
to sit for 5-10 minutes then I check to see if the large gritty bits are being
removed. If not I 180 it and start again. This way you dont have to wipe or
touch the mirror the water will do most of the work. Then with the water
flowing again I use a lens cloth dangled from my fingers with only it's own weight
and start drawing little circles from the center out. The cloth is never
pressured by you fingers or hand onto the mirror surface. The cloth literally
paraglides over the mirror surface knocking off whatever else maybe left
there.

I had an article on my method up on my website. But due to a change of
providers I'm yet to upload to the new area, beside I want to give it a lil
bit of an overhaul.

But I could send you a copy if you like.

regards