Soldering Magnet Wire
The normal insulation used in cryogenic work is Formvar, not the solder-through polyester or urethane insulation used on common magnet wire. It's also common to use high (electrical) resistance wire because the thermal conductivity of such wire is lower. Such alloys do not soft-solder well. 

Suggest you procure some fresh magnet wire with solder-through insulation if you need neither of those characteristics. Since your wire is fat you may need to heat sink the wire to control where the insulation stays.

If you need to use the Formvar insulated wire, it can be removed by special abrasive stripping machines or by very hot caustic chemicals (again, you can buy such an apparatus). You still may need to weld rather than solder if the metal is not copper, or use a more active flux. In the latter case, be sure to clean thoroughly.

Another approach would be to use bare nice shiny pre-tinned wire and add insulation in the form of heat shrink tubing, ceramic tube insulators etc.