I doubt it could last in the historical situation. The steppe empires (Avars, Bulgars, Turks, Mongols, etc) or vanished, or crumbled on their own weight when regularly defeated or being unable to lead raids that were the base of the ruler legitimacy, or assimilated themselves on a pre-existent base (Bulgars : byzantine, Mongols : China, Persia, etc.).
On their direct neighborhood, Khazars had little room to develop. Their best shoot would be to take over Russia and all Volga basin but doing that would precisely prevent the rise of Russian khaganates and Rus' kingdom.
They would more probably have to take over part of Byzantium or Persia and I doubt they would have the numbers or the organisation for that.
Did Judaism actually prospered? It seems that it was limited to a little part of Khazars elite, whom the majority kept previous rites with some being Islamized since the IX century. And their population and vassals mainly kept their religion.
I doubt you'll end with a "Jewish Khanate", even if the rulers kept it. Again, such nomadic/steppe states were relatively comprehensive on religious policy, as they were composed of many different cultures and religions (that eventually merged with them if at the same cultural/political level, kept vassals if not) they didn't have interest to take over.