Low-cost way to bullnose & polish edge of granite tile
There are dry polishing pads made for marble and other stones:

Amazon: Dry stone polishing pad set

These are made for an angle polisher that runs around 3000-4000RPM, so one of your drills would work, but you'd need a mandril adapter that you could chuck up in the drill since the pads don't have a post for the drill.

The process is to start with the roughest pad and make several passes with each pad (remove all visible scratches from the last pad before moving up to the next one), so it's a time consuming endeavor.  It takes a little skill to keep the pad moving and buff it all evenly, but it should be too much of a challenge.

The description says that for the best results, use with water.  That could be a challenge depending on your drill setup - just don't use too much water.

And for what it's worth - I've never seen this in stock in any major retail store.  It's just too specialty of a product for them to devote shelf space to.  Most people still leave any stonework to professionals.