dead wire - how to replace
 

An outlet in my bathroom is dead. Breaker is switched on. I replaced the plug just in case it was bad, but that wasn't the case. The outlet is fed from a 2 switch box (ceiling fan, and lights), but is directly spliced into the neutral and black (which is live). I tested the black wire at the outlet and it is not live. I'm guessing a drywall screw or staple, and years of moisture from the bathroom caused it to finally give up?

There's a stud between the two boxes (about 1' apart). Any ideas on how to replace this short piece of wire? Can I tape the new to the old and pull it through? or will I have to cut a hole in the drywall on either side of the stud to help fish it through?

Thanks!