Originally Posted by
mudnut
Use a lighter to burn the insulating lacquer about 5mm off the ends to be joined. Scrape the carbon off and then tin them with a solder before joining them. Solder quickly so you don't destroy any more insulation. Use a thin bit of electrical tape over the join, rewind the coil and Bob's your uncle.
I forgot to add, solder the ends side by side and bend the join to fit the curve of the coil. Twisting the wires together before soldering will make a lump in the coil.
If you need some wire and are a hoarder like me, try stripping it out of an old charger transformer or the like.