twist stripped ends together, solder and then heat shrink. Don't forget to put the heat shring on before joining the wires together and keep it well away from any heat whilst soldering or it will shrink where you don't want it to [speaking from experience ]
Braid the individual strands, then use a crimp sleeve?
I think you will not get a reliable wire without soldering. The problem isn't the electrical connection, but tension resilience.The core conductor isn't attached to the sheathing so it can (and will) move slightly. If you build the new wire by only securing the sleeve, there will always be a (minor) possibility that the conductor moves and loses contact.
Don't skimp on the overlap when you do this. 4mm should be safe.
edit: oh, and don't use metal crimp tubes unless you intend to re-wrap them in tape. Very easy to short-circuit your amp...