> after Solomon's death and centered around Ephraim. These tribes were dispersed by the Assyrians around 700 BCE.
Right. Actually, transported to northern Iraq. They subsequently dispersed themselves.
> There are some remanents who remained in Israel and intermarried with Assyrians who settled there.
Yes. There were always stragglers. Likewise, there were some from the Southern Kingdom taken captive by the Assyrians and moved NorthEastward into the Northern Kingdoms captivity and subsequent diaspora.
> These peoples practice a form of Judaism derived from that of the Northern Kingdom,
Oops. The Northern Kingdom was never Jewish, thus did not practise any form of Judaism. Israelitish maybe, but not Jewish. Their primary religion became disobedience, and their punishment was becoming "Lost" for a prescribed period of time.
Regards ... LT