What kind of ESCs are you using?
If you have Plushs they have build-in BECs and might screw your kk-board up by all submitting ~5V to your board but with a minor differance between the BECs. Sometimes the electronics might get a bit pissed off by that.
I don't know if the HK KK-board is sensitive to that, but it might be, and that would explain why it works if you only have one ESC connected. Try cutting, or somehow disconnecting if you don't want to cut it, the red wire (middle one) from your ESCs signal wire on the other two ESCs and try again. But first off, check weather you have a BESC (built-in BEC) on your ESCs.
Remember to leave at least one of the ESCs untouched, otherwise you wont get any voltage at all to your board.