Because I have to wiggle the cable around the USB end to get it working it was pretty evident that the problem was one or more broken wires within the USB plug. Hence the solution was to strip the plug apart, cut out the dodgy cable and re-solder the cable onto the plug.
Needless to say I wrote this instructable with a nice working mouse! Oh and this is my first instructable, so enjoy!
Step 1: Cut off & cut open
Next we stand the plug on its side, with the exposed metal part up in the air. The exposed metal is the bit you want to hold onto so you don't cut your fingers. Now using a stanley knife or similar to cut one side of the plug. (See the photos!) You want to use a fair amount of pressure whilst moving downwards with the knife, just please please please be careful not to cut yourself.
You want to cut until you can part the whole rubber/plastic casing and extract the metal innards. Hopefully you shouldn't have to cut the other side of the plug.