Does Apple Care cover Magic Mouse that I bought with my iMac?

Yes it is covered. It is likely just fine. Please do the following:

The following will require any USB keyboard and mouse. If you don’t have them, borrow or buy them, they are inexpensive!

1.     Connect the USB keyboard and mouse to the computer.

2.     Open System Preferences – Bluetooth

3.     Un-pair the wireless keyboard and mouse.

4.     Turn off Bluetooth

5.     Restart the computer in Safe Mode per the directions in Use safe mode on your Mac - Apple Support

6.     Restart the computer normally.

7.     Open System Preferences – Bluetooth

8.     Turn on Bluetooth

9.     Re-Pair the wireless keyboard and mouse.

10.  Unplug the USB keyboard and mouse and begin testing the wireless devices again. 

AppleCare+ is supposed to cover included accessories. Are you sure it needs service? Mice rarely go bad.

Yes.

Have you plugged the Magic Mouse into the computer lately to charge it?