If you can get the part setup, a carbide end mill will chew through a tap.
On that same note, ask your dentist if he has any samples or no longer used Dental Burrs, that he will part with. Mostly Carbide bits silver soldered on to a steel shank. If he is willing to dump those in your lap, you are laughing! Good stuff.
Put those in an air pencil grinder (60K revs, on LOTS of air), or even a Dremel tool, and you can carve your way (carefully) carve your way through a broken off HSS Tap, as well as to carve a slot to fit a screwdriver bit in a stripped out head of a screw. Lots of revs, low side pressure. Be patient, and get great results!
Done it a LOT, saved a PT-6 Engine change on a Bell 412, because of a broken HSS tap, and saved a lot of high dollar parts for the CF-18 Heads Up Display, that had a bunch of #3 size screws... Have carved my way all the way through the stub of a broken off screw and picked out the remains, too.
We had one repeat job, that was the retention hook, that held an AIM-9 on the rail in flight, which was a really common job in our shop, removing the broken off stainless steel bolts that had sheared rather than coming undone. I saved every worn or chipped 1/4 inch carbide bit, freehand sharpened them, and used them to plunge out the remains of the sheared bolt. Then the remains of the stainless Heli-coil is removed and replaced. Much money saved...