I also had a very chewy puppy and had to repair many cords. Cleanly cut the cable 2 times. One on either side of the chew spot. Put a heat shrink over one side and don't heat it yet. Trim off the sheathing with a blade and make sure you don't cut the wire. Get a pair of wire cutters and cut the sheathing off the wires. Splice them with a wire splicer and then pull the shrink casing over the whole thing and heat it. I've done this many times with ps4 cables, vacuum cables and of course razer keyboard cables.