Unfortunately, the Wifi and Bluetooth card in the laptop uses a proprietary connector so there is no way to upgrade it internally. However, you could buy a USB Bluetooth 4.0 dongle and use that. The only drawback is it will take up one of your USB ports whenever you use it.
I had a Wavebird that was having the same issue. You need to replace the stickbox. You can order an aftermarket stickbox from Amazon or Aliexpress. What I prefer to do is to get a Wii Nunchuk with a good stick and desolder the stickbox. The Nunchuk and Gamecube controller use the exact same stickbox so you can simply solder it onto your Gamecube controller.
It’s possible you may have a bent or missing pin. Look inside of the controller slot and if a pin is bet you should be able to bend it back, however if it’s missing I would try to find a new connector and solder it on.
If your simply going for functional all you should need is a new logic board and any missing brackets or screws. If you want every feature on the phone to work you will need a new rear camera, a logic board with paired home button, a main speaker and if you want the phone to still be somewhat waterproof you’ll need new adhesive for the screen.
I had a similar problem a little while ago. I would recommend letting it sit on the charger for at least a few hours. I had a phone that I thought was bricked but I let it charge overnight and it worked perfectly after that. If that doesn’t fix it you may have a bad battery in which case you should find a new one and replace it.