how to clean nasty, dirty finger prints on LPs

A blowtorch judiciously applied should do the trick.

Try pure 99% isopropyl alky if you can still find it or straight 90% if not. Put a drop on the print and lightly spread it to cover with a corner of your record brush. Let it sit a few moments. Then brush it round with the brush as usual. Try again if it doesn't work the first time, then give up. The stuff's great for removing adhesives from CD cases too. Side note: in the United Kingdom isopropyl is kept behind the pharmacy counter to prevent young people from ingesting it for fun.