After drawing the grid. I used a nail to mark the intersections where I need to drill. Then I drilled a small hole.  Here is how I envision the shift registers would be mounted on the PCB that is the base of the cube.

After drilling the hole to the same size as the LED, I used this drill bit to smooth out the rough edges. Left is before, right is after.

Still undecided where to bend the leads and solder them to the structural wires.