In article AOg9f.361787$1i.338638@pd7tw2no,
says...
I broke a tap today. There was not enough protruding for me to get a grip
and twist it out. I thought I would anneal the tap in situ and drill it
out. So I heated it red hot with my TIG torch and let it air cool.


Since you've got the TIG. weld a nut to the stub of the
tap. Depending on the tap's size and how much is sticking
out, it may be helpful to countersink the nut in order to
reach the end of the tap. If the tap is broken below the
surface or the stub is very short you can build the end up
with weld before laying the nut over it.

309 SS wire will work well, but 308 or 316 is OK, Inconel
or other high nickel rod is probably best. If you've got
some, old fashioned trichlor based tapping fluid will make
it much easier to get the tap moving.

Ned Simmons