Hello ludditebuttrying,
Thanks for posting in the Apple Support Communities.
We understand you're looking for assistance with your keyboard not connecting to your MacBook Air, right?
If we have that correct, let's first confirm if this is isolated to the keyboard or all Bluetooth devices. If you have any additional devices to pair via Bluetooth, this would be the quickest way to isolate this to your keyboard. For assistance with pairing and connecting a Bluetooth device, check out the steps found in this resource: Connect a Bluetooth device with your Mac - Apple Support
If you see this is isolated to the keyboard, try pairing the keyboard with another device if you have one available. This will further isolate what could be occurring, so check out the following resources for assistance with pairing to other devices:
Set up your Magic Keyboard, Magic Mouse, or Magic Trackpad with your Mac - Apple Support
Set up and use Magic Keyboard for iPad - Apple Support
If that isn't helpful, you may need to have the keyboard serviced. To check into this in more detail with you, contact Apple Support directly to check into your options.
Have a great day!