We just bought two Magic Mice today. Spent similar frustrating time trying to get mine to work with my iMac 27" desktop computer. First of all, the cable that came with the Mouse does not fit in my standard USB ports, it's too small. It's a USB-C cable designed to work with newer Apple laptops. When I tried plugging the Magic Mouse to my computer with a regular USB Cable, I got a message on my screen that said I should unplug the device because it was too much power to recharge the Mouse! Like you, I turned it on and off, etc. So.....then I used my old Mouse to go to System Preferences>Bluetooth where it showed me what Mouses I had, I turned my old Mouse off (and took the batteries out), then my new Mouse was highlighted, I clicked on it and it was connected! I checked the amount of charge, and it showed 99%. Hope you can find a way to get your Mouse to work.