I'm not exactly sure how you are wiring the switch, but I don't think you should need any wire nuts.
Get an extension cord and cut the female end off (or buy a wire already prepped for this) and then attach the wires to the switch. Then run wires from the switch to the outlet. You could even use a piece of the extension cord you used for the incoming power.
No need for wire nuts at the fixtures as they have their own attachment devices.
14/3 will be plenty for a router (15 amp capacity), and this is a common size for an extension cord. There is a good chance the cord on your router is 16/2.