I tapped the remaining 36 holes in the 10" CF Flange with a new tap. I raised the tapping depth by 0.0463" in case it was a chip in the bottom of the hole that broke the tap. I then added the groove pattern with the 1/8" end mill. The grooving would have hit the broken tap and possibly damaged the end mill, so I did not groove into it.
I will go to the chemistry machine shop tomorrow to see if I can use their sinker EDM to remove the tap. I will then retap or remake the hole, add the grooving pattern, then clean the two 10" CF Flanges.
I dropped off this part and some drawings at the Caltech CCE Machine Shop. They will remove it with a Plunge EDM. They say the hole will be usable.
I spoke with the shop and the part should be done by tomorrow (11/5) morning.
The CCE shop gave me back the part. The tap has been removed and I don't think I need to rethread the part.