That's really unfortunate friend.
To remove the wire you don't need a knife at all (which as you found, strongly risks damage to the conducting wire itself).
You simply dig at the very end of the memory wire with your fingernail (which will expose it through the thin heat shrink tubing they used to attach the memory wire), and then you "peel" the memory wires back like you are peeling a string cheese stick snack:
Then at the IEM end of the memory wire, sometimes you can pull the memory wire out, and other times you can't (which if you cant you just move it around in many directions to weaken it and it just snaps off).
As far as repairing your cut wire, since the insulation is the only thing cut (and not the copper conducting wire strands), I would just paint it with a few coats of "liquid electrical tape". You can purchase this at most "big box" hardware stores (Home Depot, Lowes) in the electrical section, or on Amazon/ebay. It's a flexible rubberized coating, that will be far superior to trying to wrap it with tape (it will look better too).