If you can find the correct switch, it would probably be easiest to find someone with a microscope and a small soldering iron, because it's not a difficult job.
The switch itself is just on/off. The long press / short press functions are done by one of the integrated circuits on the board. You only need something that shorts the contacts when you press the button. Disconnecting the battery will stop the device, but it might cause it to lose memory, and when you reconnect, it might not turn on until you press the switch.
In the switch like that there is a metal disc under the button. You can see the pads that it contacts in your picture with the green arrow. The disk is concave so that it only touches the pads around the edge. When the button is pressed, the disk becomes flat and connects to the middle pad, while still touching at least one of the pads around the edge.
If you buy one of the switches, you can just take the top part, the button and the metal disk, and fit it to what is in the photo. The metal part in the photo of the switch is only to keep the button and disk in place when the switch is released, so some glue or a rubber band around the whole board might work for that.