SPOILERS AHEAD
(For the Trek book Dark Mirror)
Bryan, jab1, thanks. I do recall seeing one or two of the DS9 episodes, now that you mention 'em, but it was years ago (probably when they first aired.) Unfortunately, I’m still confused. Well, that or I’m just missing some important details, here.
The following assumes that the bit I mentioned, about everything Trek is canon, is true. (If it ain’t, well, say no more, because the book I read just becomes an interesting little “what if?”)
In Dark Mirror, the mirror Starfleet (the Empire) still exists, and in fact, the Klingons, after having significant parts of their homeworld laid waste, are slaves/subjects of the Human/Vulcan Empire. The mirror Enterprise is equipped with techno-doodads that enable it to pull the real Enterprise through into the mirror dimension, way out in the middle of noplace.
Without breaking down the entire story, it turns out that a few crew members need to sneak aboard the ISS Enterprise, with the intention of laying hands on the techno-doodad, and getting the hell home.
Picard, of course, eventually has to join the gang, where he runs into the mirror Worf. The two end up having a discussion awfully similar to the one Kirk had with the mirror Spock way back when.
(In a rather neat little sidetrack, Picard accesses the Imperial records on Spock, and learns that he did in fact try to reign in the Empire, only to end up the victim of political assassination.)
So, according to this, the Empire is still the dominant faction, the Klingons are a slave race, and I don’t think any mention is made of the Romlulans. In fact, the reason for this whole scheme is that the Empire needs new worlds to conquer, fast, or it’ll fall in on itself. Is this how it was explained in DS9, or did I just read a neat story? 
Incidentally, the book mentions that the story takes place during the 4th season of TNG. That have any bearing on anything?