I was at the Milwaukee Makerspace tonight and brought the razor to repair. It took me about an hour to do the work but I think I fixed it.
There is only one screw holding the case together. After removing the screw, the case comes apart with a bit of coaxing.
One of the wires on the power cable connects directly to the power switch. The other wire is connected to a wire from the razor via a crimp connector. The first step was to de-solder the wire from the power switch. I could not find any de-soldering wick so I just applied some heat and forced the wire off.
Rather than cut the wires going into the connector and shortening the one from the razor, I used a vise to expand the connector and remove the wires. That was surprisingly easy.
After disconnecting the power cable from the razor, I cut the wire just below where I thought the break in the cable was. Next, I stripped about one inch off of the end of each cable. The existing cable was tied in a knot that looked like a pretzel. I tied the end of the remaining cable into a similar knot.
The final steps were to reconnect the individual wires to the switch and the razor. I soldered one wire to the switch. The other wire was connected to the wire from the razor. I was able to use the original connector and re-crimp it.
I tested the razor and it appears to work! Looks like I saved $30 by fixing it myself at the Makerspace.