It took me a while before I realized I'd forgotten one part… the remote.
I thought that I had a solution because we do have a TiVo/DVD-Player at the new apartment already and you can use normal TiVo remotes with multiple different TiVos so I thought I'd just change the code on the remote and I'd be saved. No such luck. Apparently the TiVo/DVD remotes are completely different from the normal TiVo remotes so I'm headed back to the old apartment.