Hi Airpirate,
I've never broken a tap myself, but know a few people who have ! I rescued one job by using a diamond core drill and removing the broken tap and a plug of the parent metal. I turned a plug a thou over size and hydraulically pressed it into the hole. This was then re-drilled and tapped.
I did a similar repair on a broken cylinder head stud, but using a diamond core drill not quite the same size as the stud. That one left the threads in the hole which were removed with an easy out after blasting some heat at it.
So using a diamond drill might be your answer.