Reddit mentions of 10 Pcs 1x40 Pin 2.54mm Pitch Straight Single Row PCB Female Headers

1. I used 22 AWG solid core wire. Which should be plenty. Check that the wire you're using isn't thinner than the wire that's already there and you should be fine.
   
2. Just one wire to each contact. The purpose is to just make the switch short the two wires together, so orientation does not matter.
   
3. For mine there are two positive and negative labelled POW+ and POW-. Again orientation doesn't matter. These are the type of pins I'd use, they should simply snap off to however many you need, just solder the wire to the metal pin and plug them onto the motherboard.
   
4. The way I've done mine the power button on my case does not work anymore. However if I used new header pins I would solder the original switch wires to the header pins as well. Theoretically this should allow either switch to operate as normal.
   
5. ...
   
6. Profit! ^No ^house ^fires!
   

You could go with something like THESE and cut them to the length you need, but they require soldering and heat shrink tubing to be applied...