Need help fixing broken wires

Those wires usually have an enamel insulating layer... so soldering will not take quick as easily as normal wire.
You can solder it back together just fine, but you may need to 'burn' off some of the enamel, I do this by melting a small ball of solder and keeping the end of the wire submerged until the solder begins to stick. Keep in mind that the solder wont be insulated! So use electrical tape or better yet, heatshrink tubing to isolate each of the wires.
And no... you do not have to separate each and every strand from the wire and solder independently. Of course, unless you really really want to

 

I recommend cutting outer insulation jacket back about 5 inches or so on both cables, so you can stagger the slices. Therefore they won't all be bunched together.

 

Could also gently scrape the varnish layer off with a knife.

Another avenue would be to replace the cable and resolder it inside.

 

I've done it in the past, but yes, you need a lot of patience.
If these are high end headphones, I would order the right cable.