Hi E_C_H_O,
You can boot into the Surface UEFI to check if the Type Cover will be responsive. To boot into the UEFI, refer to this page. If it does not work in the UEFI, then it means that what you're having is a hardware problem. I would recommend that you take the device to a Microsoft Store or contact Microsoft Support for assistance.
If the Type Cover works in the UEFI, then the problem is related to the software. Please run the Surface Diagnostic Toolkit to check for any errors. To download the tool, open this link.
In case you need further assistance, please reply to this post.
Regards.