Perhaps part of the reason is that numerically, by tribe count and also general population (and especially future population), the 10 tribes way out-number the Southern Kingdom. Given the fact that Judah is called “more treacherous” than her sister Israel, it makes sense that Yahweh would have divorced ALL Israel had He not had promises to keep (for the sake of David and Moshiach, and, I believe, the relative protection of His Torah). So really, the only reason why all 12 tribes weren’t “divorced” is because he need to keep a remnant together. In this light, then, the title “House of Israel” and the Northern Kingdom represents what should have happened to all Israel, because of disobedience and whoredom. It might be His way of letting all Israel know that the One that was “only” sent to the “Lost sheep of the House of Israel” was ultimately to be the salvation of “the WHOLE house of Israel”; the House of Jacob. Does this make sense? Just my personal thoughts.