OK, so, rather than buy and install the new cable, I figured I'd give it a try and patch it up. Didn't work. So, some of the wires were undamaged. A few had some of the insulation/coating scraped off, and the white wire was completely worn through the wires.
So, I wrapped the lightly damaged wires with electrical tape.
And "patched" the white wire as pictured below. The tail lights/running lights work. But, the signals and brakes didn't work. Any thoughts about what to check next?