Does that wheel assembly even contain the switch? Every mouse I’ve repaired has had a small SMD tact switch on the PCB for the middle button.

To find a replacement switch, I go to Digikey or Mouser, filter by the approximate size, then look through the pictures until I find something that looks right. Then I download the datasheet and check the exact dimensions to see if it will fit. Tact switches are cheap, so order a bunch. I usually have to replace them every few years. If they are available with different actuation forces, get several different ones to try out.