If you decide to have another go at cleaning this mirror...
I am a little concerned about the spots left after your first cleaning. Hopefully, they will clean off. Those are my favorite kind of spots! You might consider soaking it for a bit in a water and soap solution, then proceed with fingertips and ROR procedure. Rinse with distilled water.
If this doesn't remove the spots, the coating on the mirror may be etched. The only remedy I know of then would be to have it re-coated.
I hope the mirror still performs sufficiently. I am not a fan of shipping mirrors for re-coating, but is necessary sometimes.
Good Luck!
- Phyllis