It's an unbelievably thin stranded wire, but the wires are coated so they can be in contact with each other without shorting. It's all twisted around a thin thread of cotton or nylon to add strength, then then encased in it's sheath.

The trick is to gently scrape the stranded wire with a blade for the solder to stick and to make a good connection.

I've repaired a few headphone wires; theyre usually thin copper wires covered with enamel insulation. Burn off the insulation with a blob of solder, or sand it off, and the solder will stick.

You’re right, the kind of cable often used is not easy to solder. This makes it hard to solder a broken cable together again, or to replace a broken / bent plug. So best replace the entire cable and its plug — it’s still an inexpensive part.

You’ll need to solder it to the contacts inside the can, but that’s quite straightforward.

In case the internal cable that goes from one can to the other breaks, you can replace it with any bit of audio cable so you can use one that’s easy to solder.