Insulation on the wire will compete with the LEDs visually...
You're going to need a mile of this stuff; make your segments as long as possible.
Rosin core solder will work a treat... or you can use a flux pen.
Follow your template as you solder, making sure that each joint is strong and doesn't bridge any of the LED legs.
Soldering on the outside gives you a little more space and can help prevent short circuits.
Time to solder the lines that drive your cube: power, ground, and signal-in.
Pass those connections through one of the posts and into the base.
Add a bit of hot glue to each of the corners / posts.
At this point you may want to solder a temporary connection to the cube, and run a 'strandtest' to make sure that everything is hooked-up correctly.