In case anyone wonders how this turned out ... I could not get the 'drill out' tool to work. The first problem was getting the hole centered in the broken bolt. I might have had better luck if I had ground it flat first, as international3414 suggested. The drilling end of the tool didn't drill into the bolt very well. The bolt metal is very hard, as someone else had mentioned. Using another bit, I was able to drill a hole, but the drill out end of the bit just did not seem to grab well enough to turn the broken bolt out.
I was able to replace the entire 'nut', as it's called, and the problem was solved, but I had wanted to save the original part as the color is slightly darker on the new part. Now that I have the nut off of the guitar, I still think I will be able to turn the broken bolt out of it. It will just take a different approach. I think the drill out tool might have worked had I been able to get the hole centered in the first place. With an off center hole, the drill out tool just couldn't apply pressure evenly.
I'm happy to report that I did end up with a perfectly repaired guitar and there were no injuries :)