If it's a diagonal, I shake it like a thermometer to get most of the water off. You now have a clean mirror. Normally, as far as a vacuum-deposited reflective optical coating less than a micron thick is concerned, if something is not necessary, it is best to refrain from doing it. A good guideline is to never clean the mirror and then proceed from there. This can extend the life of the coating.
Alcohol is fine.
Just before Step 4, with the mirror still at a steep angle, slowly and carefully pour alcohol down the face of the mirror. Remember alcohol is a solvent. Do not get any on the back of the mirror as it will erase or smear any markings or QC stamps there. The trick is to get FULL coverage on the front yet don't get any on the back. I use an absorbent washcloth as a dam at the top and the baster to squeeze alcohol on the extreme upper portion of the mirror, and then pour from the bottle to get the rest.