When splicing wires together, it is important to get an electrical connection that will keep conducting even after your soldering iron has cooled off. Paramount to this is getting a good mechanical connection between the wires, not just an electrical one.
I was able to use TechShop's soldering stations and helping hands to keep everything in place while I worked.
Freshly stripped multi-filament wires need some attention before they will take well to being soldered.
First, the filaments of each wire should be twisted together, or pigtailed, to keep them orderly and behave more like a single entity.
I prefer to add some soldering resin to the filaments at this point too. It really helps the solder flow when tinning the wires. If using flux core solder, this step isn't critical, but it still does help.