If the point is to fight the Klingons, then why is mission completion based on the Borg side of the area??
It is actually a race between the Federation and Klingons to complete the mission first. If the other side complete it first you get a telling off by your bosses and no phat loot as a reward.
But as the Klingons have a war zone all to themselves, the sensible ones go there to do the anti-brog mission without any interference from the Federation. So the only Klingons you will see in Ker'rat are there for some PvP fun. Assuming they can be bothered to leave the exploration clusters.