I thought putting glue in the part that didn't break off and trying to put them together and pull it out would work, but now it's just glued in there.
Both times, you were drunk, weren't you?
Acetone will dissolve super glue, but unfortunately it dissolves or damages about every plastic. However somebody recommended this for use on plastic models:
換気扇レンジクリーナー
made by リンレイ.
Google translation says that's "Ventilation fan oven cleaner", made by "Linley."
Somebody else mentioned using 2% sodium hydroxide (lye) oven cleaner, but that can be really nasty on metals.
Apply acetone with a Q-tip or eye dropper while hold the socket upside down, and try to pull out the stub with a drill bit or tap. And because you're you, do not then use that Q-tip in your ear or that eye dropper in your eye.
It's probably going to be easier to just open the iPod and unsolder the socket.