Hey there,
Just had a thought about this. Reading through, your saying the drivers are not there, visible, for the keyboard?
When you plug your keyboard into other devices, they work. But not on your device. Now, I maybe wrong with this idea, But it is worth a shot.
Have you tried downloading the essential drivers for Surface 4 here:
https://www.microsoft.com/en-us/download/details.aspx?id=49498
Regards,