Eventually brute force and impatience took over, this is what it looks like:

It came apart pretty clean. The two brass pieces hold the cartridge at the two plastic rings and make the contacts, the white part is a spacer between the two. The third wire is soldered to a cable that disappears inside the handle. Like I assumed the cable broke at the spot where it used to get warm, but it was a little more inside so I couldn't see it just by removing the rubber part. Cut the part of the wires off, soldered it back on, figured out how it goes together again, et voilĂ , working iron

. Really didn't think it would be that easy.
I opened up the other one with the blockage inside as well. It is constructed a bit differently, there are three metal pins that are fixed to a piece of plastic that makes up the whole diameter. They broke off clean from that surface, so fixing them will be tricky. Also I can't really get through from that side. I tried to push it out from the other side with a cartridge, but it didn't move. But now the cartridge sits as deep as it is supposed to... Will have to resolder the three pins and see if it makes proper contact or if something is still blocking inside.