Step 3: Wire Everything Up
Strip the wires and solder each pair to one LED. E.g., solder the solid brown to the anode of the farthest LED on one side and the brown/white wire to that cathode of the same LED. Then use orange for the next LED, green for the next, and blue for the next. This isn't set in stone, but just makes it easier to keep track when you're connecting the other ends of the wires. Repeat for the LEDs on the other side. You've got two unconnected LEDs left, right? The two on either side of the center hole. Use the third hunk of CAT5 to connect the remaining LEDS (pick whatever color pairs that you like).
Solder the solid color wires to the 5-pin connectors in the same order as they are soldered to the LEDs, using one connector for the LEDs on one side of the lid. I don't recommend soldering with the pins in the Arduio - you could damage the on-board connectors.
Now solder the striped wires together into one big blob (well, something a little neater than that), and then solder a 2" - 3" piece of wire to that. This jumper wire should be stripped on both ends, since the unsoldered end is going to be plugged into the GND socket.
Now you should be able to plug one connector into pins 2-3-4-5-6 and the other into pins 13-12-11-10-9 with the brown wires connected to pins 2 and 13. Plug the jumper wire into the GND pin at one end of the digital I/O connectors.