FAST6191 will give you a run for your money when it comes to Xbox related things.

 

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.

 

Sorry my description yesterday was a bit vague - when you remove the faceplate of the 360 there is a green cable that runs from the faceplate to the 360 its this one that i have broken....

 

My comments from yesterday still stand but ribbon cables tend to want to be replaced more than they want to be fixed. Have you damaged the connector at all or just the ribbon?
I have yet to break/replace one so I am not sure about sources and I just watched a video on replacement and it does not seem too bad if you have otherwise pulled enough of the 360 apart to get at the faceplate.