What happened, after Attila's death, to the Huns is relatively poorly researched (or rather, not researched at all). I can only name two historians who have dealt with the 6th century Huns, and a third mentions them in passing while dealing with the Sclavs (the "ancestors" [not really] of the Slavs, who fought the Huns a lot). In terms of historical sources, they pop in and out of the picture. Hērnäk (Reconstructed spelling. Ernakh by Greek reckoning, i.e Priscus'), Attila's third son, apparently killed his brothers, and probably inherited most of Attila's wealth. Attila died very suddenly at a feast, according to the Romans, and a power struggle started immediately between the three brothers. Priscus also mentions how Hērnäk was his father's favourite son. Later on we meet Hērnäk's descendant "Goto", but we don't know how he's related to Hérnak, only that they were of the same Hunnic clan. He caused trouble with the Byzantines and got crucified, that's about as much as we know of him. I can only imagine Hērnäk inherited it, his sons split it, and so on and so forth. The Huns were basically trapped in the Balkans and part of Asia Minor, too weak to break out and with nowhere to migrate. One only has to observe their petty raids, barely mentioned in historical sources, until finally being overwhelmed by both the Magyars and Türks. They were likely then subsumed into the general population.

Fun fact: Hērnäk probably means "Lucky Little Man." Reconstructions of Hunnic/Hunnish vary wildly, but this is one of the most academically sound ones, I forget which author wrote it though (will find it though, I own the book). As is the consensus, it deals with Hunnic as a language in between Mongolian and Turkic, closer to the latter than the former, but still very different.