I use a Dremel tool with an abrasive/diamond bit for the hardest stuff.
cut
The tool can used to cut out a broken tap too.

You mean grind............How do you grind a 1/4x20
tap a 1/2 inch long in a hole with a Dremel without
grinding the bores threads off. You must have a lot
of time and an extremely steady/accurate hand.

Additionally how do you get the center of the abrasive
tip to grind when you plunge, seeing how it's surface
speed is minimal.