Another idea for how to fix this: This is a bit risky and very tricky, and only might work, but is less invasive than properly isolating and repairing the individual wires.

The part of the cord that's broken contains 2 even tinier wires. It's possible that these 2 tinier wires are themselves intact (except for the insulation) and are just shorting out on each other. IF this is the case, you may be able to get away with isolating the 2 tinier wires and then casting them in epoxy. You'd probably need some kind of tiny hook-like tool and a magnifier sufficient to work with the tiny individual strands. If both conductors are unbroken and you are able to physically separate the strands, that ear bud should start working again. If successful, the area could be cast in epoxy to add some insulation and rigidity.