I've just watched the scene, and if you look closely (when you can see his face in the mirror) you'll see that there's a strange texture on his head right around where he is about to drill. That is probably some piece of prosthetics / make up so he could drill into that without drilling directly into his own head.
It's also possible that he angled the drill in a way that it doesn't go perpendicularly into his head or the drill bit was a bit loose in the chuck so it's actually going into the drill instead of into his head. Different tricks could have been used.
The ambulance would be necessary in case he drilled too far into the prosthetics and ended up hurting himself, so that was a real risk.