I am not really sure what there is to say specifically pertaining to the xbox, generally if you snap/cut a wire then best to join it back up or solder a new length (if you pulled/fatigued it then consider this more). If you broke it from the board see if you can solder it back in else you get to solder it to the trace or the place it would eventually end up as well as anywhere the now broken pad/trace went to (with the xbox I imagine you would be spared a lot of this but if the wire came in an branched out in two directions then things get more tricky, or not depending upon where the branch happens).
You can still buy replacement parts but with original xboxes not exactly going for a lot these days you might consider those instead.