I find it very interesting that there are no Kinnos for the Exile and Disappearence of the Ten Tribes. There are plenty for the loss of the Temples, for the massacres that took place during the seiges of Jerusalem, for the loss of Torah scholars and even for the communities of Europe that occured during the Crusades and the men, women and children who were martyred in them. All of these horrible events deserve kinnos.
But there is barely any mention of the fact that before the First Temple was destroyed, over 85% of the Jewish nation was taken away in chains. Surely there were massacres during the seige of Shomron (Samaria). Surely many of them died on the way to exile. And, the fact remains, that they are missing from Judaism to this very day. The Great Mosaic of Israel has been reduced to shreds, with only two sections of the original twelve still extant. Whether one wants to hold to the legendary "they're living beyond the Sambatyon" answer or the more mundane "they intermarried and are gone" answer, the fact remains that they are no longer here. Imagine how much fuller Judaism would be today with the influence of the other ten tribes for the last 2500 years.
It is true that there are some passing references to the Ten Tribes in Kinnos, but not many. The most explicit would be the Kinnah "Shomron" which is recited at Tisha B'Av night. However, even this mention of the TT is only made in conjunction to what happened in Judah/Benjamin.
So, why is this? Why are there no Kinnos for the vast majority of Jews that disappeared from the pages of history?
Is it possible that it is because of the rivalry that existed between Judah and Israel? I don't think so. It's been thousands of years since this rivalry existed. Surely in that time, someone could have come up with a Kinnah.
Is it simply because they are gone from the pages of history but we (the remenants from the kingdom of Judah) are still here? I'm not so certain about that either. After all, the kinnos themselves are all about things that we no longer have - the Temple, the Service, the Torah scholars and communities that have been wiped out?
I suppose one could say that it is simply because they have been reduced to a few sentences in II Kings which describes their exile. Unlike the periods of the destruction of both Temples, there are no extensive records from the Jewish point of view of the destruction of Samaria and the exile of the people -- and as such, there was little "source material" to work with.
But, I would still think that someone, in the last two thousand years, would have come up with something...