I would say there is a 99% chance the problem is inside the strain relief gland where the cable leaves the volume control on the headset side. To repair it will be difficult but not impossible. You need to open the volume control unit, I can't see in Logitech's pictures how it fits together, if it has screws it will be easy, if it is welded you might be able to open it by squeezing gently in a vice until it cracks along the joint. Don't blame me if it doesn't come apart cleanly!
When you have it opened, remove the strain relief gland (the bendy bit with slots around it) and note exactly where the wires in the cable are soldered to the board. Unsolder the wires and snip off a few cm of cable so the bad area is isolated. Now, try to remove the wire from the strain relief, you might be able to pull it out but usually the molding process makes it bond to the outer layer of the cable. If all else fails, grip the relief in a vice and run a small drill through it to remove the original wires.
Next, thread the cut end of the cable back into the strain relief, it isn't vital that it fits tightly but it helps. A tiny amount of lubricant might be needed. Then trim back the insulation to expose the wires inside the cable and re-solder them back to their original places.
Basically, you are cutting out the bad bit and rejoining the cable again. It will be a few cm shorter but that isn't normally a problem. If you do replace the whole cable you need to note how many wires are inside it and it's outside diameter so you can buy a replacement. It shouldn't be hard to find a suitable alternative cable but you might find it is only available in big and quite expensive reels.
Brian.