The nose was broken off when I pulled the starter. First, what could have caused it and second, what's the best way to get the broken piece out?

 

Id say the only thing that could have really caused that is if the starter was loose when it the truck was being started. The torque from the starter could have cause it to jolt in one direction breaking the nose cone off. The the rest of the starter nose is back where the flywheel is, then pull the inspection plate off the back lower part of the engine and dig out as much as you can get to. Thats about the best you can do with the engine/trans connected.

If you get a new starter and the truck turns over just fine with no funny noises coming from the bellhousing, I wouldn't worry about any pieces being in there. Chances are they probably fell out when the starter broke anyways.