I did that once in a piece of stainless and a local specialist used the equivalent of a reverse arc welder to remove the broken off tap. He called it a tip eroder. It wasn't cheap but it worked.

Another time I did that with a tap on my prop. I was trying to add a place for a grease fitting. Older maxprops had to be disassembled to grease them but newer ones have 6 mm setscrews that you remove and put a grease nipples in their place. For that one I just put the prop together and a year later, on the next haulout the tap had rusted away.

I have had very bad luck with taps breaking. Now I am paranoid about backing them out after every 1/4 turn. It takes a lot longer but saves time in the long run.