I have a recently purchased Surface Pro 11. I also had this problem. Following the suggestions of some other respondents,
Turning off the computer with the keyboard or touchscreen, rather than by closing the keyboard hinge mechanism, seems to avoid the problem.
I have had at least 8 successful instances of starting or waking the computer with no keyboard problem.
Gordon in Dublin, Ohio
Try turning the computer off using your method and immediately fold the keyboard/cover over the screen. Does it start the next day with the keyboard working? What I have found is delaying folding the keyboard over the screen for a few minutes will have the keyboard responding the next day. I haven't checked it lately but if you fold the keyboard over the screen immediately AND turn the computer back on within the next hour (not over night) the keyboard will respond. Yes, it doesn't make sense. It is like the computer is doing something although it is turned off.