The House of Israel is specifically the Northern Tribes ( Northern Kingdom) of Israel:
Asher, Gad, Naphtali... etc (All but Judah and Benjamin, which assimilated into Judah/Jews).
The house of Israel were taken into Assyrian captivity. The tribe of Judah was taken into Babylonian captivity. ( See Kings, Chronicles )
Judah came out after 70 years and tried to recover and continue their heritage and customs ( See Ezra, Nehemiah )
The House of Israel did not come back and were scattered and assimilated into wherever they went.
No one knows who they are (in terms of bloodline) ...
So, Jesus came for many things ( atone for our sins, become the model of how to live etc.)
But He also came to regather the lost tribes back to the way of life God had commanded.
Jeremiah Prophecy
Then I will gather the remnant of my flock out of all the countries where I have driven them, and I will bring them back to their fold, and they shall be fruitful and multiply. 4 I will set shepherds over them who will care for them, and they shall fear no more, nor be dismayed, neither shall any be missing, declares the Lord.
5 “Behold, the days are coming, declares the Lord, when I will raise up for David a righteous Branch, and he shall reign as king and deal wisely, and shall execute justice and righteousness in the land. 6 In his days Judah will be saved, and Israel will dwell securely. And this is the name by which he will be called: ‘The Lord is our righteousness.’
7 “Therefore, behold, the days are coming, declares the Lord, when they shall no longer say, ‘As the Lord lives who brought up the people of Israel out of the land of Egypt,’ 8 but ‘As the Lord lives who brought up and led the offspring of the
house of Israel out of the north country and out of all the countries where he[
a] had driven them.’ Then they shall dwell in their own land.” -Jeremiah 23:3-8