Identify the receiver first
Logitech uses several receiver systems. Unifying receivers are marked with the orange Unifying symbol. Logi Bolt and LIGHTSPEED are different and are not interchangeable with Unifying.
Connect the Unifying receiver
Plug it into a USB port. A device that was paired with another receiver will not necessarily reconnect automatically.
Use Logitech pairing software
The older Logitech Unifying Software was designed to add and remove Unifying devices. Logitech no longer maintains it and recommends Logi Options+ for supported products, while older hardware may still require the legacy tool.
Power-cycle the device
The pairing flow normally asks you to turn the mouse or keyboard off and back on so the receiver can discover it. Confirm the connection when movement or a key press is detected.
Test after pairing
After pairing a mouse, check clicks, wheel and side buttons. For a keyboard, test every key. This helps confirm that the original problem really was the receiver.