Hi John,
Welcome to Microsoft Community, We apologize you are getting this problem on your Surface pro 11 cover keyboard. Possible reason might be Hardware connection issue, Physical damage or defect. Thank you for sharing the prior troubleshooting steps you made by updating the drivers and been through restart. We understand the frustration that you are getting to this problem but don't worry we are here to help you.
We will do step by step troubleshooting to fix this problem.
Please follow this steps:
DISCONNECT / RECONNECT KEYBOARD or TYPE COVER
Disconnect your Surface Pro 11 from Keyboard or Type cover.
Check for Debris that blocking your Surface and Type cover connection.
Reconnect your Keyboard or Type cover on your Surface Pro 11.
If still Keyboard or Type cover lights did not turn on or did not work submit a service order.
please follow steps below.
SUBMIT SERVICE ORDER
Open any browsers
go to account.microsoft.com/devices or
https://account.microsoft.com/devices/select-device-for-repair
register or log in using your Microsoft Account
Start Service order
Provide all the information on your device Surface and Keyboard or Type cover Serial number, Model, Etc.
If there’s anything more you’d like to share or questions, please don’t hesitate to reach out.
Warm regards,
Jansher Liwag