How to repair basic cable/wire issue for household appliances
There's no good way to repair just that wire, and the adjacent wires are also probably close to failure. You'll need to remove a section of cable and rebuild. 


Label the cable a ways from this end so you know the orientation with respect to the white plug. As you can see, the cable could be reversed and have the same color scheme. You don't want that or your fade and balance will be backward (or worse). 
Snip all wires as far from the plug as you reasonably can. I'd try at the start of the thicker shrink wrap. The more wire you retain the easier repair will be. Strip the wires 3/8" and set the plug aside. 
Cut the cable 3-4" back from the strain relief--wherever you think you have conductors in good condition. 
Salvage the strain relief, or install a new one. I'd try pulling it off the cable, but it's probably thermally welded in place. You can possibly pull he copper out of all chases, slice off the cable jacket flush on each end of the part, and use a drill to reopen the passages. If you can't save the entire thing, just save the solid portion that secures the cable in the speaker housing and use that. 
Feed the fresh cable through the strain relief. Put a square knot in the two cable bundles inside the strain relief block to prevent pullout. 
Begin stripping wires. Notice that the outer two cable bundles contain two wires each. That'll be some fiddly work. Be patient and don't damage the conductors. 
Strip the wires 3/8" and slip appropriate heat-shrink tubing over each. 
One at a time, twist the wire ends together parallel and solder them, then apply the heat-shrink tubing. 
Connect the plug and assemble the speaker housing.