Add me to the list of those who've not actually heard of anyone breaking that cord. The clip maybe, but the cord is pretty tough - were you pulling at it with a tool like pliers?
Anyway, the replacement isn't too hard. You have to remove the glovebox again, then remove the entire panel the glovebox attaches to. The damper cartridge is attached to the back side of that panel. You can replace the cartridge with a little finesse, or just replace the whole panel.
I have a spare if you want it.