For this mod you just need a Qi Receiver. I recommend getting a USBC Qi Receiver intended for android devices. These can be had off of ebay (at least in the USA) for $3~5 shipped. For some reason I did not have luck with an iphone variant. The USBC version is nice because you can check to see if it works to your satisfaction before physically modding anything.
Do NOT get a generic DIY qi receiver (available on amazon and ebay for about $10) as pictured below. These tend to emit a high pitch squeal when charging.
Cut off the edges of the paper with a utility knife and soak the entire thing in isopropyl alcohol for a few minutes. This should allow the paper/sticker parts to be peeled off, revealing the coil, ferrite backing, and circuitboard. Do not remove the paper on the ferrite backing, it just creates a mess.
Desolder the USBC connector, it's a cool low profile right angle connector that you could use to charge unmodified Game and Watches on display.
In order to make room for the Qi receiver circuitboard, the inside of the back casing needs to be slightly modified. Using nippy cutters, just chop off the regions of plastic shown below.
To get it to fit in the game and watch, I extended the wires connecting the receiver coil to the circuitboard. I then taped down the coil and ferrite backing to the center of the game and watch.
See image below for areas to connect ground and the 5V coming from the receiver. Alternative to soldering ground, you can just attach it to any of the mounting screws anywhere (this is what I did).
Make sure to protect against any potential shorts with kapton tape.