A broken tap can be castrophy
as they are made of very hard steel harder than most drill bits but:
1] Walton Company makes a broken tap exractor which has long fingers and is inserted into the grooves of the tap (makes sure you know if you need a three or four finger extractor] Sources can be found on net.
2] bring to machine shop and have them remove tap with plasma cutter
3] bring to machine shop and have them remove tap with 2 to 5 mixture of nitric acid and water. If you do it, be careful. This often just loosens tap.
4] Put dry ice into rubbing alcohol in a thermos bottle and then put several drops of -20 degree alcohol onto the broken tap with a dropper and strike tap smartly and extract pieces. Be careful with dry ice.
5] drill hole in broken tap with a diamond bit; in hole insert square extractor.
6] wheld a nut onto tap and turn nut. Keep welding nut onto tap and turning until broken tap in rmoved.
I hope others post the methods they have tried.