I'm thinking the only place that a survivable Jewish kingdom might be possible would be someplace in the British Isles, and if it is going to have a good chance of surviving, we'd basically be talking England.
There is a possibility, albeit small, that this could have happened in much the same manner that Kievan Rus supposedly went Christian.
Let us suppose that, about the year 600 A.D., one of the more powerful of the kings of the pagan Anglo-Saxons...let's say
Redwald of East Anglia, for sake of argument...is perhaps a bit more worldly than he was in OTL and is aware of the three Abrahamic religions, and not just of Christianity. He asks for representatives of the Christians, the Muslims, and the Jews to come and make cases for their respective religions.
The Jewish representative regales the King with tales of Joshua and especially David, and makes the case that Judaism is a warrior's religion. He presents circumcision as a blood sacrifice to God...something a pagan Anglo-Saxon would understand and identify with. He also slyly intimates that it could be seen as the mark of a warrior, a man who is not afraid to face his worst fears. Any man so marked, he argues, whether his trade was in warfare or not, was a warrior in the eyes of God.
These things appeal to the King, and instead of converting to Christianity as per OTL, he decides to convert to Judaism. Using the same arguments the Jewish scholar had used, he persuades most of his noblemen over the next few years to follow suit, and under their guidance, the people of East Anglia convert over the next ten years. Furthermore, when Edwin of Deira seeks refuge with him a few years later, Edwin converts to Judaism as well. When King Aethelfrith of Northumbria tries to force Redwald to turn Edwin over to him, Redwald, as per OTL, raises a large army, marches north, defeats and kills Aethelfrith, installing Edwin as King of Northumbria. Edwin proceeds to convert the Northumbrians, with Redwald's help, over the next few years.
Together, Redwald and Edwin attack and subdue the Mercians, the Lindisware, and other Anglo-Saxon and British tribes. Redwald and Edwin, sometimes using the carrot, and other times the stick, begin converting the populations of those regions to Judaism. By the time Redwald dies c. 625, all of England north of the Thames is either Jewish, or in the process of becoming Jewish. In Southern England, Kent is Christian, and Wessex and Sussex remain pagan.
Let's assume that this alternate Redwald has no son, or his sons are killed in the campaigns described above. Let's assume he does have a marriageable daughter, and he gives that daughter to Edwin. When Redwald dies, Edwin becomes King over all of England north of the Thames.
Let's also assume that Jews from all over Europe, at the invitation of Edwin, have come to make their homes in his kingdom. With the help of Jewish scholars, he is able to organize his kingdom better and more efficiently than his rivals to the South. His tax base is managed more efficiently, and he's able to consolidate his rule over the conquered lands, enabling him to further move along the conversion of the population to Judaism.
This stronger Edwin, who has already disposed of the Mercian royal line and added Mercia's power to his own, easily defeats Cadwallon ap Cadfan if that ruler of Gwynedd tries making trouble for him (since his OTL ally, Penda of Mercia, is gone, he probably doesn't do that anyway). Edwin thus lives well beyond the time he died in OTL, finally dying in, let's say, the year 650.
Edwin's eldest son by his East Anglian bride...let's call him Osfrith...inherits a strong, well-organized kingdom whose population is majority Jewish and whose non-Jewish population is rapidly declining as a result of an ongoing program of conversion including taxes levied on non-Jews and various other rewards to those who convert. He finishes what his father and grandfather started, crossing the Thames and defeating the remaining kingdoms there...first Kent, then Sussex, then Wessex. The other kingdoms are unable to band together to oppose Osfrith, and are defeated separately and incorporated into Osfrith's kingdom. By the year 670, all of Britain, from the Firth of Forth to the Isle of Wight, and from the borders of Wales to the English channel, are in the hands of Osfrith. He passes this along to his successors, who continue to consolidate their kingdoms and convert the populations to Judaism. By the year 700, the conversion process is complete, and England is a unified, strong Jewish kingdom.