We'll start by preparing the wires. If you used a computer power cable like me, the first thing you'll want to do is cut the ends off, strip away about an inch of outer insulation, then about a 1/4 inch of individual insulation. You should also strip the ground cable (ignore the contradictory image note). Do the same thing to both ends of the cable. For now, we will only be using one end of the 115-volt cable. On that end, apply a generous amount of solder to tin it. Then insert the wires into the barrel connectors on the key switch. While heating up the connectors with your soldering iron, apply solder to the wires where they contact the connectors. Also, attach the ground wire to the switch's metal casing. If something fails, this will protect you from electrical shock.
For the LED wires, strip about 3/8 in. off one end of the set, and about 3/16 off the other. On the long side, solder on your LED, but not before you slide heat-shrink tubing down. I forget to do that every single time.