The western union splice is rather awkward in stranded wire. If you flux the wires and lay the ends parallel before applying any solder, then put a tight whipping (seizing) round them with very thin tinned or bright copper wire, you gain nearly as much mechanical strength as the western union splice. A single strand stripped from one of the wires in the dead lead would probably be suitable.
Alternatively, go to Maplin, buy the crimps and if you don't have the proper tool and can't be arsed to buy a compound action ratchet crimper, strip the hard blue or red plastic, squeeze the crimp onto the wire with pliers and solder it, with a wet rag round the wire to chill it so the solder doesn't wick up it, and feeding solder into the end of the wire from the terminal end of the crimp. Then you *MUST* heatshrink over the crimp.