We've come to the part where you ruin a perfectly good lamp dimmer.
Go ahead. Cut that dimmer off the end of the cord.
You can get rid of the dimmer. We are not going to use it.
Now, strip the ends of the wire and solder it onto the switch. Use the center pin and one of the side pins on the switch. It doesn't matter which side pin you use.
Don't forget to slide the shrink tube onto the wire before you solder!
I used three sizes of shrink tube I had laying around. Your selection may vary. What we want to do is provide some electrical separation and strain relief for the wires and terminals.
I used small shrink tube first on each wire, then larger shrink tube to cover the smaller shrink tube and then finally big shrink tube to cover part of the switch and the other layers of shrink tube.