I have had to fix many pairs of headphones before but almost always encounter the same problem with the type of cable used:
It is so difficult to tin them and heating them too much always results in the thin plastic outer layer (jacket) melting.
It feels like the wire has plastic or other fibrous material woven with it to increase it's tensile strength and keep it kink free.
Anyone have tips on the best way to solder this type of wire?
Cut the wire
Strip of the rubber insulation
Cut away the strand of nylon, leaving two enammelled wies per channel
Heat the wires with a soldering iron, keeping the flux flowing
Tin the wires by adding solder
You're done!
I used to use a solder pot for this, but in the interests of health and safety, the easiest way is to use some liquid flux, or maybe a flux pen, and just brush the bare copper end, a quick heat with a clean, heavily tinned iron is all that is needed for the solder to flow from the iron tip to the wire...