Bluetooth

Try resetting the BT module. Assuming the keyboard and mouse are Apple devices (e.g. Apple Magic Keyboard / Apple Magic Mouse) try the following.

First obtain a spare wired keyboard and mouse, and reset the iMac's Bluetooth module. To do that hold a shift key and an option key (two fingers) while selecting the Bluetooth icon in the Mac's menu bar. Three additional options will appear; one to reset the BT module and another to factory reset all connected Apple devices. Try one, and then the other.

The third option removes all devices, which I doubt will help.

To reiterate, you don't want to do that unless you have wired USB input devices available. They don't have to be Apple branded — anything will work.

You will subsequently need to re-pair the keyboard and mouse; write back if you need help.

For reference: If you can't connect your Magic Keyboard, Magic Mouse, or Magic Trackpad to your Mac - Apple Support