Step 3: Put the U-HID in the Button
Find a way that the whole U-HID assembly will fit inside the button, then use some hot glue or silicone to stick it in place.
Replace the mezzanine, metal spring, and PCB. Now comes the soldering. The U-HID Nano is going to just sit there and wait for pin 10 to touch the ground pin. We're going to use the normally-open momentary switch inside the Easy Button to interrupt that connection, so the circuit is only closed while the Easy Button is depressed. Solder the gray and black wires as shown.
It would be a good idea to test it at this point to make sure it works before you put everything back together. Use a little hot glue to make sure the wires do not interfere with the holes in the PCB. Those are what keep the button aligned properly, and if they're blocked, you can't push the button.