Hi Alex,
Welcome to Microsoft Community.
I'm Hahn and I'm here to help you with your concern.
I'm sorry to hear that your Microsoft Bluetooth Keyboard is not working properly. It sounds like you have tried some of the common troubleshooting steps, such as replacing the batteries, checking the Bluetooth settings, and restarting the Bluetooth Support Service. However, it seems that the keyboard is not entering the pairing mode, which could indicate a hardware issue with the device.
According to the Microsoft Support website [Troubleshoot problems with your Microsoft mouse or keyboard - Microsoft Support], if the pairing light flashes three times, then turns off, your keyboard might have a problem that can be fixed by updating the firmware. To do this, you need to download and install the Microsoft Mouse and Keyboard Center, and then follow the instructions to update the firmware of your keyboard. This might help to resolve the issue and make your keyboard work again.
If updating the firmware does not help, or if the pairing light does not flash at all, then your keyboard might be defective or damaged. In that case, you might need to contact Microsoft Support or the retailer where you bought the keyboard to request a replacement or a refund. Contact Us - Microsoft Support
I hope this information helps you to fix your keyboard problem. If you have any other questions, please feel free to ask me.
Your Sincerely
Hahn - MSFT | Microsoft Community Support Specialist