I started out wanting to retrofit an existing stick, but then scrapped that idea and just used the hardware in a wooden case.
I’m getting the power from the Neutrik USB port next to the HDMI on top of the case. It’s USB B on the outside and USB A on the inside. I then have a short USB A to Micro USB cable on the inside giving power to the pi.
I made a board on the inside that has all the button connectors from the original stick mounted on some breadboard and then uses a ribbon cable to connect to the GPIO pins on the pi. All the buttons have four wires because of the LEDs so keeping the existing connectors was easiest.