Thanks for your reply. There seems to be no direct way to reset your keyboard except by removing and reinserting the battery, which you have already done. It seems like you have tried everything with your keyboard, including testing it on another device. Since your mouse is working just fine, it is likely a hardware issue with the keyboard itself.
Regarding the warranty, as far as I know, Microsoft only offers a 1-year warranty. I understand that you can't locate the serial number, which is needed to check the warranty on the Microsoft website. https://support.microsoft.com/en-us/warranty
You may try contacting Microsoft billing support to double-check: https://support.microsoft.com/contactus/
I'm sorry that you are locked out of your PC right now. Usually, you can re-enter your password again after 2 hours. You may use the On-Screen Keyboard for this.
From the login screen, click the 'Ease of Access' icon beside the power icon. Select 'On-Screen Keyboard' from the menu, and then use the On-Screen Keyboard. Thank you.