Left hand drill set, nice find.
If this were my problem, I'd first soak the threads (let them stand overnight) with something like a teflon spray, or liquid wrench. Then, preferrably with the caliper off the car, I'd heat up the caliper, and try using the left hand drill. I'd try a size that was less than the drill size for the threads on the bleeder, but with enough bite to dig into the bleeder. I don't think you'll be able to get the bleeder to turn out, since you couldn't do it when it had a hex on it. But it is worth a try.
When this didn't work, I'd follow one of two courses.
My biggest concern with this first approach is that it could screw up to bottom of the hole where the bleeder seats. For this approach I'd use a regular drill, one size less than the drill size for the bleeder threads and drill out the hole to the bleeder. Then I'd follow with the drill size for the tap for theese threads, taking care not to oversize the hole (which is why I'd start with one size below this first). Then I'd follow this up with a normal tap, and then a bottom tap.
The other approach which I think could work better is mentioned in previous posts, putting in a helicoil. I assume the drill for the helicoil for the stock thread is larger than the current threads that are in the caliper. If so, you can drill it down to just past where the stock threads are in the hole. So once you are past the threaded portion of the broken off bleeder, the rest could just be pulled out. This should protect the bottom seating area for the bleeder. Helicoil it, and you are back in business.