My two cents worth. Now I have read all these possible solutions and even tried most of them removing or at least trying to remove broken taps, with little or no success. I out of desperation got one of these EDMs to reduce this very frustration a Chinese one at that, but it does work. The biggest tap, a blind hole 16mm tap in S/S. Using a rod the diameter of the center web of the tap, it just erroded the center of the tap and the three flutes just then fell out. Messy with the residue, but component undamaged, which is all that we are after. I have even drilled holes in torx bits as one customer ordered several thousand with no holes in the end.
My toy drilling, as can be seen very messy and also not so fast, but the only way to do these as the ends are hardened. I have done about 500 of these very boring.
A Walton tap extractor might be worth looking at.
It's a three or four pronged type of thing, and they are usefully effective.
Cromwell used to stock them, or you could try Rotagrip in Brum ?