How do you think the Empire was created and what made Cochran kill the Vulcan's
Possibly something as small as the Mirror Universe's U.S. Constitutional Convention deciding that they wanted to call their head of state "king" rather than "president". No change in powers, just the difference in title, might have been enough over the course of time.
But who knows? The only thing obvious is that it had to be something fairly fundamental, yet minor enough that it didn't cause such sweeping changes that any of the main characters from TOS were never born or were born with different names or even for there to be a dramatic difference in ship design.
As an aside, I tend to think that Cochrane and Sloan knowing about the Borg and telling the Vulcans is the point of divergence between the Prime and Nu universes - but I've gotten in quite a few disagreements about that on the board.