I have had repeat and random issues with my iMac and my original bluetooth keyboard and mouse. If I plug the mouse it says 'charging' but unplugged it will not stay connected. Keyboard will only run plugged in, even though it says connected.

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

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• Connect the USB keyboard and mouse to the computer.
• Open System Preferences – Bluetooth
• Un-pair the wireless keyboard and mouse.
• Turn off Bluetooth
• Restart the computer in Safe Mode per the directions in https://support.apple.com/en-us/HT201262
• Restart the computer normally.
• Open System Preferences – Bluetooth
• Turn on Bluetooth     
• Re-Pair the wireless keyboard and mouse.
• Unplug the USB keyboard and mouse and begin testing the wireless devices again.

I have a mouse and keyboard that seem to drop out and are very slow in operation

Please read the thread and follow the steps I already noted above.