Removing a tap which has been broken off below the surface of the workpiece can be done using the following procedure:
If the hole provides clearance, a coated electrode can be used to build up the tap until it is above the surface. This is best accomplished by dipping it in and out of the puddle. The coating on the electrode helps to prevent damage to the threaded walls of the hole.
Once the tap has been built up above the surface, the end can be filed into a square, gripped by a wrench, and the tap backed out.