Today we had a locksmith come out and remove the remainder. I was worried about getting an adequate replacement for the doorknob, since I've had repeated problems with not being able to find acceptable replacements for defunct products. The first couple knobs the locksmith offered were the crappy modern cylinder kind. But after I described my parameters, he produced a knob very similar to the original. \o/ He also moved the latch to a better position so the door will catch easily instead of having to be manhandled into place which is how the old knob got so beat up.
Problem solved. :D