As far as I know, you don't put the wire through the holes. Here's how I do it:
Clamp the items you want to solder into something, you need to hold the iron and solder and only have two hands.
put some solder on the prongs, the hole will fill up with solder giving it a strong connection to the prong. cut the plastic off the wire and twits the copper ends a bit (twist them in the direction the rest of the wire core ) to make them nice and tidy.
Put a small amount of solder on the wire ends too. You want the complete outside covered, but do it as thin as possible, not a solder lump.
I always prefer to work fast, melting the solder quick and apply. If I do it slow I always end up with a messy connection, and it won't be as strong either for some reason. And you may end up heating up the switch and melting something.
then clamp one end to hold it in place, and puth the wire where you want it to be. Touch the prong with the iron untill the solder on it melts and let the wire melt into it. Because you already aplied solder to both ends you probably don't need to add more.
It's pretty easy with a little practice.