If you do try washing it you could try isopropyl alcohol, which you can get from Jaycar <http://www.jaycar.com.au/productView.asp?ID=NA1066>.
Oils from your skin are soluble in it and with gentle squirting should just wash off without you having to touch the mirror with anything else. It evaporates fast so make sure you drain it all off into a dish. You can mix it 50/50 with distilled water to slow the evaporation. This is a common method for cleaning camera lenses.
Cheers,
Ross...
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