I had success recently scraping the insulation off the general purpose form of this kind of cable (ie, plain ribbon) by scraping across the conductors with the tip of a sharp craft knife. It was easier and quicker and more accurate than using abrasive. I suppose there's an increased risk of causing more damage however.
I was going to suggest 0.1mm magnet wire (cheap on eBay) for the repair, but rjenkins' suggestion is arguably easier. Magnet wire is great if you're used to it...
Hot tip! When you finally get to actually soldering the wires on, tape everything down before soldering. Kapton (or the cheap version, Koptan) tape is ideal, but you could use maskng tape. Sellotape/Scotch tape is stable, but melts too easily (unless you have the very old cellulose stuff). I wouldn't use electrical tape, it has too much adhesive on it and is too stretchy. Tape the ribbon to the bench, tape your repair wires to the ribbon. When you've soldered one end, tape down the end you soldered before soldering the other end (because the solder will melt again and the wire will move and your beautiful craftsmanship instantly turns into a botched mess. I speak from experience!). If you feel you can safely cut the wires after they''re soldered in place, it's easier to use long strands and trim them after soldering. (But it's more important to not do more damage, obviously)
There are at least a couple of YouTube videos showing repair of these cables, might be worth a look.