I have just got myself a NUC also in the past week or so and have been searching for this problem that the original poster and many others have described in this thread.
My NUC specs are:
D34010WYK (Bios 0033)
8GB Corsair Vengeance 1866MHz DDR3L @ 1600MHz
128GB SanDisk mSATA
And most importantly a Logitech K830 Keyboard
I first thought it was interference from something in my living room where i use the NUC. But i don't have anything in there to really interfere with it. Just a 46" TV that the NUC is connected to via HDMI and a Samsung Soundbar that i haven't actually had on since getting the NUC.
I can bearly get 5' away from my NUC with the K830 before getting intermittent signal. I then tried the K830 on my main PC and it works perfect at any distance within the room i have it in. Whether facing the PC, away from it or even in the next room. So there is no problem as far as I am concerned with the keyboard itself.
I am going to investigate some more tomorrow in regards to disabling the IR sensor and anything else I can think of to (Although I shouldn't really have to do all this considering the price of these NUCS). Don't get me wrong. If it wasn't for this pretty big problem then it would be worth every penny.
I just hope this issue can be recognised and be replicated as a real issue by yourselves at Intel and for there to be a solution soon.
EDIT: Sorry I forgot to add that I am fully up to date with all drivers also.
EDIT 2: Well I have been testing with my NUC today and at first i disabled the IR within the bios. There was no change in the keyboard behaviour.
I then got out the USB extender cable that came with the keyboard and run it underneath my TV cabinet with some tie bases and attached with some small cable ties (It looks very tidy and can't be seen) and the actual dongle is just at the front left hand side barely visible and low and behold it all works perfect now. I can sit back at full range on my sofa with it working as intended. Finally!
So in conclusion I would say that there is some sort of interference from the NUC's metal casing it would seem. Or at least in my case anyway. I know some people have tried this without success but other's have and it as worked like it as for myself.
To add to all this as well and what some people have mentioned about there not being enough power from the USB's. I run my K830 keyboard, a Logitech webcam, an external USB powered 500GB Seagate HDD and a XBOX 360 controller all at the same time off my NUC and with this new keyboard setup there are zero problems now. I also have it all running through my Samsung soundbar via HDMI now which as a wireless Sub.