I'm sure I'm making this harder than it needs to be, but it took several iterations to make the "vertical" jig.
I probably should make even more so I don't have to keep moving them as I solder.

I got impatient and increased the print speed on my printer and that lowered the print quality.  Note how the LED is held crooked.  Thankfully, I was able to straighten it out by holding soldering iron near the trouble spot and correct the hole by inserting an LED to reform the hole.

With six jigs, the structure is finally rigid enough for soldering.  Here's one finished wall.  Horizontal wires are anodes that will be common per layer.  Vertical wires are cathodes (eight per wall, 64 total).  You can see the excess vertical wires at the top of this photo.  These will elevate the lowest layer 4cm above the base, so the cube would appear to be floating rather than resting on the base.