Right at the end of the chapter, the elevator completely breaks and I can't progress. I'd really like to know what's happening here.

The elevator breaks if you enter it too fast, you have to wait for the door to fully open before stepping in. If it breaks, save at the save station next to the elevator and reload your file. The elevator will start back up at Level K but it will eventually come back down.

So it's breaking because I'm speedrunning too well. Nice.

that is a good thing because you know you going fast.