Most of the problem around this issue is that medieval Judaism is essentially made of scattered communities : not only as a factual situation, but also as the basic structure of Judaic life.
For a Judaic state to appear in Europe, you'd need at least a coherent enough regional structure, which didn't existed and didn't looked like it was about to exist.
You'd argue that Khazars seems to have partially embraced Judaism or at least a variant of it, but it's more of a case of a really limited part of the ruling people (you don't have much archeological evidence, mostly litterary and onomastics) within a super-complex chiefdom structure, than a Judaic state as nowhere at any point Khazars seems to have either attempted a up-down diffusuion of Judaism or a religious-political objective.
Even a Judaic community somehow managing to carve for itself a piece of territory, or even a Jewish individual being in a clear position of power will be eventuallt (and rather quickly) crushed by neighbours : probably much less by feudal and islamic states' leaders or religious heads, but more probably by mobs lead by knights (
miles until the XIIth were the lowest bar of nobility at best) and small nobles. I'd point to you the pogroms of
Rhineland in 1096 and the
slaughter of Grenada of 1066 for what matter mobs' threat and how were percieved Jews being in position of power above non-Jews.
Your best chance, and that's a stretch, would be to buff some Jewish communities in mediterranean basin, allowing some of their ruler to participe in the general administration of some regions on their own.
Al-Andalus' fitna seems to be a better situation for this to appear, with a Jewish micro-taifa : but how long would it last before Christian expansion, Taifas expansion and Berber dynasties expansion (especially these, which weren't exactly Judaic friendly to say the least, at the point you could witness Jewish migration into Christian Spain)? I don't think it would last more than some year as an independent entity, and not that much longer as a relatively autonomous community under a Christian or Muslim overlordship.
And that's assuming there would be a community sucidal enough to decide it's a good opportunity in first place.
Other, but even less independent, attempts would be to buff the IOTL medieval Jewish dynasties.
For exemple, the
Kalonymoi of Narbonne, which beared the title of
Naci (prince) and claimed a davidic ascendency, ruled over the Judaic community of the city (and maybe countryside as well) being nicknamed, altough it might be anachronical, "Kings of Jews of Narbonne"
and certainly had an actually autonomous authority. They weren't the only Judaic nasi, but seems to have been one of the more prevalent and more lasting ones.
You could, with tweaking hard enough history (for exemple, maintaining a coastal Byzantine control in South Italy, where Kaonymoi seems to have came; or
by maintaining the Carolingian policy toward nacim; or yet by preventing the feudal desintegration of the region that alienated a lot of Kalonymoi land estates)) manage to create subordinated Judaic dynasties more or less integrated into palatial organisation and/or regional autonomous existence (on communities, tough, not territorial). Safe that, which I admit doesn't fit your OP, I don't see much alternatives.
(EDIT : Kalonymoi are but one exemple, if a telling one : the cases of Abraham Bar Khiya of Barcelona, acknowledged as prince of all Jews of Catalonia and Aragon; or the various nacims of Rome are others)