I have a similar problem.
I did a factory reset (ESC+Q+Fn) on my DK61 PRO and my bluetooth connectivity stopped working. I can no longer pair the keyboard with any of my devices over bluetooth.
I know this is a problem with the DK61 itself since I have a Logitech K810 that I have as a backup that pairs fine with all my devices/OSes (iOS, Mac, Windows, and Linux).
Bluetooth issues appear to be rampant in the forums, and I'm hoping that an engineer from Dierya/Kemove is looking at these problems more carefully. I'm not sure what's being done about it, since the DK61 PRO isn't getting a lot of attention compared to some of the other products...
Is there a firmware update that fixes bluetooth bugs for the DK61 PRO?