Before attempting to clean the glassless mirror, use compressed air (i.e. Dust Off®, which can be found in most photographic or office supply stores) to make sure all dust and debris are off the mirror.
Use an optical grade 100% cotton cleaning cloth (no lint) and a window cleaner (i.e. Windex®). Apply Windex® to cloth until cloth is damp (saturated, but not dripping). Clean the mirror in one direction, then wipe using a fresh part of the cloth after each pass.
Handle by holding along edges only. (When holding the mirror along the edge, do not press down on the mirror surface with your finger nails.) Do not put any weight or object on the mirror surface in any position.