The “Ten Lost Tribes” are not lost. There was enough mixing of all the tribes into Judah, and, while some still traced their patrilineage to tribes other than Judah into at least the first century, common era, all of the tribes of Israel are whom is meant by “the Jews” in the Christian writings. The use of “Israel”, also, is merely an alternate reference to the same people, with an additional flavor of nostalgia.
2nd Chronicles, chapter 15, does not give an account of battles between Judah, under the reign of King Asa, and Ephraim, before mentioning that Asa had captured some of territory of the latter (v. 8). [Author note: I consider this to be an egregious omission.] The context of this mention was the purifying of religion by Asa, upon which, in v. 9,
Then he assembled all Judah and Benjamin and the people from Ephraim, Manasseh and Simeon who had settled among them, for large numbers had come over to him from Israel when they saw that the Lord his God was with him.
The successor to Asa as king, his son Jehosaphat, is recorded as saying to his ally, the king of the northern tribes, called “Israel” [as though Judah was not part of Israel], in chapter 17, v. 3,
Jehoshaphat replied, “I am as you are, and my people as your people; we will join you in the war.”
Was this merely diplomatic hyperbole, or was there truth in that, because of mixing of the northern tribes into Judah? I submit that it is the latter, and that this phenomenon continued.
In 2nd Kings 17:6 seems to indicate that all the Israelites in the northern kingdom were captured and exiled by the Assyrian, the text does not say, “all”, and the records of the Assyrians indicate that they only carried off about 27,000. Some of the people of the northern kingdom were left in the land, and some had fled to Judah ahead of the conquest.
Even by late first century, common era, among the Jews in Judaea, there seemed to have been some identification with the tribes previously north of the territory of Judah. In Luke, chapter 2, v. 36, there was a prophetess named Anna, who identified with the tribe of Asher.
Even though tribal identification was patrilineal in that time, we, now having a perspective outside of misogynist, patriarchy, know that ancestry goes back through both parents. The descendants of the northern tribes surely intermarried with the Judaeans, and are, genetically assimilated into the “Jews”.