Use silicone wire, or you'll have to cut up and resolder the chain like I did (above.)
This idea by Noe and Pedro is fantastic, they explain the procedure in their tutorial below.
The V2 connector for the current NeoPixel Mini PCB pack didn't fit mine once the leads were soldered, so I made an adjustment to their design. The V3 is below, and will fit in the caps with no problems.
Affix all of the bottom portions of the arcade buttons into the top panel, once you've removed the button and plunger.
For soldering to the NeoPixel PCBs, I used a bit of Blutack. It does get tackier when hot, so make sure you don't overheat the pads, or you'll have to use some Blutack to remove residue. But this is a huge time saver, instead of awkwardly using a third hand, or the overkill of a Panavise.
Applying a bit of solder to the pads make them much easier to solder wires to.
When feeding in the LEDs, start with the 6th LED in the bottom row, and then take your time feeing counter clockwise. It takes a bit of wrangling. Once all the LEDs are in and fed through, put in the top caps.