Steps I would take for Bluetooth keyboard connection issues:
- Check if correct profile (My logi can connect to 3 devices)
- Can it connect to the other ones?
- Batteries
- No key stuck shortcutting it from working
- Not using its own dongle
- Factory reset and try to pair
Factory reset most likely use a light on the keyboard to indicate successful, so can thus also give clue to if broken if not successful.