If the 1204 sack doesn't happen, then I think the Byzantines would be able to bounce back. Greece and the Anatolian coasts (Black Sea, Aegean, and Med) was some pretty good real estate. The problem was leadership, which would have to go to someone other than the Angelos. I think that with the way the Empire was heading, the overthrow of the Alexios III was probably going to happen whether or not the Fourth Crusade did the deed.
The Empire, under new management, bounces back under a competent military leader. The Second Bulgarian Empire is brought back under Byzantine control, and the east is strengthened against further attacks. The new Emperor is going to have to deal with a much stronger local nobility, but with the wealth of Constantinople and the annexation of the Second Bulgarian Empire, I think he would be able to reduce their power to more manageable levels.
The Empire's new management probably starts sometime in the early 13th century. If the Mongols arrive on time, this Dynasty might be able to take advantage of the reordered power structure in the East and permanently retake the interior of Anatolia.
The Crusader States' continued difficulties could end up being a boon for the Byzantine Empire. With the fall of the final Crusader cities, the Byzantines could recruit the now city-less population for resettlement on the Anatolian frontier. The Eastern Franks were very easternized, and would probably fit in relatively well to the Byzantine power structure. Re-settle them in some newly conquered areas of Anatolia, where their pretensions to independence will be of good use. These Franks have generations of ruling over Muslim populations under their belts, along with raiding and familiarity with seige warfare. They will end up being important marcher lords for the Byzantines.