How did Robert manage to keep the seven kingdoms together following his rebellion? The 7Ks only became united under the Targaryen dynasty, so once that ended, wouldn’t it be more likely for them to break apart again after the Mad King’s demise? Why didn’t everyone suddenly start declaring independence again, instead of waiting until after Robert’s death?

There’s a couple different reasons:

1. The Southron Ambitions/Rebel Alliance: Robert had already won the allegiance of the North, the Vale, the Riverlands, and the Stormlands. That’s half the continent right there, and even more so when Robert marries Cersei and gets the Westerlands as well. So it wouldn’t be seven kingdoms again, it would be one massive kingdom and a bunch of smaller states trying to avoid reconquest. 
2. Institutional Inertia: The Seven Kingdoms have been part of one consolidated monarchy for almost three hundred years, less so for Dorne. That’s three hundred years with very few wars by historical standards, internal trade, law and order, all that good stuff. So people are going to be somewhat wary of going back to being warring kingdoms, especially after a rather bloody civil war which they lost.
3. Targaryen Legitimism: even if the rest of the Seven Kingdoms wanted to start something, independence is unlikely to be the case because the Targaryens are still kicking around in Essos – as long as their kinsmen are available to be restored to the Iron Throne, the Martells aren’t about to declare themselves independent again.
4. Internal Politics: This leaves the Reach, and they’re probably the most viable candidate for an independent state given their large army and economy, although it wouldn’t be easy trying to defend themselves on the Westerlands, Riverlands, Crownlands, and Stormlands borders at the same time. But the problem is that the Tyrell claim on Highgarden runs through the Targaryens – if the Reach isn’t part of Aegon’s realm, that claim isn’t good any more and the “Oakhearts of Old Oak, the Florents of Brightwater Keep, the Rowans of Goldengrove, the Peakes of Starpike, and the Redwynes of the Arbor” have every reason to claim the Oakenseat for themselves.