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**8There were two houses after the split, the house of Judah, composed of the tribe of Judah and Benjamin, and the house of Israel, composed of the rest. The house of Israel is pretty clearly specified.***

The term "house of Israel" is often used interchangably for both groups. Also, many from the Northern kingdom moved south into Judah when the northern kingdom turned to idolatry.

Ezek 37

15 ¶ The word of the LORD came again unto me, saying,

16 Moreover, thou son of man, take thee one stick, and write upon it, For Judah, and for the children of Israel his companions: then take another stick, and write upon it, For Joseph, the stick of Ephraim, and for all the house of Israel his companions:

17 And join them one to another into one stick; and they shall become one in thine hand.

18 ¶ And when the children of thy people shall speak unto thee, saying, Wilt thou not shew us what thou meanest by these?

19 Say unto them, Thus saith the Lord GOD; Behold, I will take the stick of Joseph, which is in the hand of Ephraim, and the tribes of Israel his fellows, and will put them with him, even with the stick of Judah, and make them one stick, and they shall be one in mine hand.

Also, many from the Northern kingdom moved south into Judah when the northern kingdom turned to idolatry.

Yes, they did, that would be why there are references to other tribes in the predominately Judah remnant during the New Testament. But the Northern house was still population heavy compared the Judah.

I suppose the "house of Israel" could be used collectively, after all, they all were Israel, and Matthew 15:24 would tend to support that.

"But he answered and said, I am not sent but unto the lost sheep of the house of Israel."

But by the same token He would have included the greatest mass of Israel, for they were indeed lost, and there are indications that one or more of the disciples in addition to Paul were up in Europe. Jesus was sent, as He said, not to just to Judah and Benjamin, but all of Israel. I'd doubt that the location of the rest was hidden from Jesus.

There is also evidence of remnants of the remaining house of Israel in Europe and Assyrian records that indicate that direction for the remnant after Assyria was defeated. They would have had three/five hundred years to get there.

I don't think the chapters of any of the four Gospels are in chronological order, so it's probable that when Jesus instructed the apostles and charged them to go teach, it was referring to their life mission, not just a trip out to the towns in the region.