For the record, Komplete Kontrol has always "worked" with a computer keyboard and mouse. You can audition sounds using the arrow keys, change parameters using your mouse, etc. This new version of KK adds a LOT to this functionality by letting you use MIDI learn the knobs and the buttons. (You do this by clicking the button that says MIDI at the top)
Still, there's no way to get the full benefit of Komplete Kontrol without an NI keyboard because there is two-way communication between the software and the keyboard. You can see what you're doing on your NI keyboard, and you can't get that with MIDI learn.
To use Komplete Kontrol with the knobs on another controller, you'll need to click the knob icon on the top. That's the only way you can see the parameters that people with NI controllers see on their keyboards. And you'll be able to see general descriptions for the other pages. Even though I have a KK controller, I find having this on top to be useful to move quickly through instruments that have a lot of pages.