Tired of Mighty Apple charging you beaucoup $$$ for those badly designed power adapters which break all the time? Repair it yourself!
Step 1: Slide back the sheath
My sheath was cracked already, so it was easy to slide back.
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
Cut the power wire a couple inches shy of the strain relief. Desolder and discard the short section and the strain relief.
Step 3: Prepare wire
Strip off the sheath to about 15mm from the end.
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
Cut a very small "sock" of heat shrink tubing, and place it over the copper wire. Heat to shrink, using a cigarette lighter, a heat gun, or the edge of your soldering iron.
Step 5: Prepare plug
Desolder the old wires from the circuit board on the back of the plug. MAKE SURE TO NOTE WHICH WIRE WAS ATTACHED TO WHICH PAD!
Now, apply some nice fresh solder to the two pads you liberated, to get them ready for their new wires.
Step 6: Solder wires
First, make sure the connector sheath is STILL ON THE CORD. If it's not, slip it over the end of the cord before proceeding. I have made this mistake probably 200 times in my life :(
Carefully solder each wire to its respective solder pad.
Step 7: Sheath and strain relief
Slide the sheath back down the cable onto the jack assembly. Yours should snap into place. Mine is broken, so no snap-action here...
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 :)