Not sure what the issue was with drilling, but usual cause of failure is not having the correct drill bit (need one for metal, not wood), starting with a bit too large or otherwise trying to drill on a surface that causes the bit to wander).
I’ve found success with the following method:
1. Go to the local hardware store and buy a screw-extractor. It’s basically a spiral bit that digs deeper as it’s turned counterclockwise.
2. Back at the bike, heat the screw up a little and then spray it with WD40 or some other penetrating oil.
3. Centerpunch a small divot in the center of the screw shaft — this will help provide a start for the drill bit. (You can probably use a thick nail for this.)
4. Start with a very small drill bit made for drilling metal and create a starter hole — (Metal drill bits are meant to cut slowly — so keep your drill on a slow speed. This also helps avoid breaking the bit off in the screw!)
5. Follow with a drill bit that matches the size indicated on the screw-extractor you bought — again, drill slowly.
6. Now insert the extractor and slowly extract the dead screw.
Best of luck