Finally weighing in on this issue.
I have exactly the same problem with my Surface Pro 3.
After putting this on Reddit, Johnny redirected me to this thread.
I did some extra testing and what he says is correct.
So to quote myself:
"A month ago my type cover stopped working. Happened lots of times, so no problem, soft reset to uefi and normally it's good to go. However, not this time. Tried all common solutions but nothing. Full reset to the win10 image didn't help either. Even tried reset to the win8.1 image to be absolutely sure.
I tried a friend's type cover, same issue. He tried mine and it worked flawlessly on his surface."
On top of this, the battery drains extremely fast when in sleep mode, so these 2 issues are related to each other.
If I let my surface drain completely, beyond the level that the white "battery empty" icon flashes when you try to turn it on;
and then I connect the keyboard, plug in the charger and power on the device; the keyboard works flawlessly.
However, when I close the lid or when I disconnect the keyboard, I can't get it working again without letting the battery drain untill it is completely empty again.
I can reproduce this again and again. Works everytime when done like this.
I do hope that Microsoft takes it responsibility with this issue.
My surface is worthless now altough it never fell and was treated with utmost care.
This problem arose from one day to another and is clearly some type of hardware issue which should be covered by a warranty extension plan.
Microsoft, please deal with this issue and take your responsibility to replace faulty devices.
P.S.: I live in Belgium