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Cheers!
That's all really helpful.
I bought some of this 26awg
http://www.ebay.co.uk/itm/131824299044
which from a quick search seems to be the same as 0.14mm²I did wonder about replacing the whole wire, but I'm not sure 1) how reusable those those connectors are and 2) how easy it will be to access the space.
I've committed to the project (partly just because), but fwiw I've also bought a replacement box from Alix for less than ¼ of the cost of the bits.
This stuff should work fine. You may find it easiest to replace the entire length of wire rather than splice in a new section. If you do need to splice it in, try something like this
for the splice (and then solder that together), slip over the heat shrink and you're all sorted. Don't bother with tape, it'll go manky. If you don't have any wire stripped then a carefully wielded penknife is normally sufficient.
As for soldering, remember to tin the tip of the soldering iron - clean it on some damp sponge/kitchen roll and then dab some solder on the end. This will help the heat to transfer onto the wires. Then hold the tip against the wires to solder and after a second feed in the solder. It should wick into the strands. Keep the heat on for as little time as possible to avoid melting the insulation.