Lots of good ideas.

I've found tap extractors usually don't hold up very well. When we break a
tap, the first thing we do is head for the dull carbide bin. If you are
careful, you usually can cut them out with a solid carbide endmill. On a
very rigid milling machine. The tap is going to break into pieces, and those
pieces are going to chip and ruin your carbide, so we keep some dull ones
around just for this kind of problem. Carbide is very resistant to heat, so
when the cutting edges are nearly gone, crank up the RPM and apply some
pressure. Resist the urge to use coolant. When the tap starts to glow (and
maybe the carbide too), it will also start to cut. If you're not the type to
wear safety glasses, better start now! Something WILL break. This almost
always works if the tap is large enough to use and endmill. If not, we go to
the EDM.

BTW, the corners just break off of a flat bottom. We usually try to use a
ball endmill.