Update: I got hold of a pretty decent tapping head. I tried it out today. It took a while to get the hang of it... I underestimated how much pressure you need to put on it going down. I wasn't helping through enough to start, and that was kind of screwing things up. But once I got the hang of it, I got through about 100 pieces no problem. It sounds stupid, but I actually just made a little holder out of a piece of wood, then two little pieces of wood glued on top to fit the round in-between. The whole thing can slide around and adjust, plus the sides hold them loose enough that they can adjust. I backed the torque off until it was disengaging, then put it a little above that. Unfortunately it doesn't look like I will be able to use the combo drill/taps on this drill, as there is too much run-out. But maybe on a different one some day. But I'm just happy to stop breaking taps at this point.
But anyway... Knock on wood, it looks like it's working. Thanks everybody for the help.