There is an acquaintance of mine that used a really small end mill - in a drill press - to drill out the end of a broken off HSS tap - successive passes with his x-y vice until he got break through at one of the holes along the tap - it more or less fell out at that point - maybe want to try with an end mill, instead of a twist drill bit? Something telling me it was a tungsten carbide cutter on end of that end mill - maybe from Amazon.ca?? It was really small - I do not remember if that was a 6-48 tap or 8-40 - but he did not want to mess up the tap hole, nor the threads that he had cut already - just using a "dull" tap, I guess.