Xbox 360 Wireless Controller's dongle not working for Windows 10

The "Xbox 360 Wireless Gaming Receiver for Windows" (product page) is prone to blowing the F1 fuse (seen here), after which it stops working completely. The reason the fuse fails is thought to be from overheating.

If your receiver has failed in this way, it can be fixed by replacing the faulty fuse. If you don't have a replacement fuse or aren't comfortable soldering surface mount components, you can try bridging the contacts with solder as a temporary fix. In general it's not safe to bypass a fuse in this way and this could result in damage to the receiver or the host it's plugged into.

Thanks to ICptJackSparrow for his repair guide, this is where I first learned of the issue and how to fix it:

https://replayproject.wordpress.com/2010/10/09/how-to-repair-microsoft-xbox-360™-wireless-gaming-receiver-for-windows®/

iFixit also has an excellent guide for this repair:

https://www.ifixit.com/Guide/Microsoft+Xbox+360+Wireless+Gaming+Receiver+fuse+replacement/118338

If you choose to bypass the fuse rather than replace it, keep in mind that the receiver is now at greater risk of overheating and should be considered a potential fire hazard. Make sure it gets plenty of air flow and avoid leaving it plugged in when not in use.