Step 1: Slide back the sheath
Assuming yours is intact, I think a good swift twist/tug with a pair of pliers should snap it loose.
Step 2: Cut off faulty wire
Step 3: Prepare wire
Separate the copper strands from the green nylon, from the smaller wire.
Twist the copper strands.
Cut off the green nylon.
Strip the smaller wire.
Tin the ends of both wires.
Step 4: Heat shrink tubing
Step 5: Prepare plug
Now, apply some nice fresh solder to the two pads you liberated, to get them ready for their new wires.
Step 6: Solder wires
Carefully solder each wire to its respective solder pad.
Step 7: Sheath and strain relief
Since mine was broken, I applied some electrical tape to bandage it.
Now for the coup-de-grace: Heat up your glue gun, and fashion a nice strain relief out of hot-melt glue!
Et voila! Just as good as new. Actually, probably better than new. And: it's got that post-apocalyptic look I know you love :)