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If you have the broken extractor out (never use those by the way) drill the remaining broken bolt out, oversize if necessary and use a bottoming tap instead of a starting/tapered tap. No need to go deeper in the original hole. If you stop just short of drilling out the original threads you may be able to use the original size tap and cut out the threads of the bolt with the tap.
Bottoming tap is straight sided with very little taper (only about 1 or 2 threads)
Thanks Mflat
I cant get out the broken extractor/bolt. I did pick up a 5/16" bottoming tap this morning at Grainger.
I dont know exactly how deep I can go, so I'll start at 3/16" deep and hope the newly tapped hole will grip the new bolt.