bluetooth

The following will require any USB keyboard an 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 https://support.apple.com/en-us/HT201262

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. 

How to Reset Your Mac's Bluetooth Module to Fix Connection Issues...

https://www.macrumors.com/how-to/reset-mac-bluetooth-module/

What all USB things are connected especially Bus powered items... USB, Wifi, & Bluetooth all share power from the same thing.

You can take it to an Apple Authorized Service Provider that has the correct diagnostic tools. Do any other Blue Tooth devices pair to the computer? For example wireless earphones or headsets. If they do, then Blueooth on the Mac is fine. Either you made an error on the instructions above OR you may have a defective keyboard.