The answers already provided are more than adequate, in my opinion. My contribution here is not for their improvement.
In my view, our starting point for understanding and evaluating the Old Covenant kingdom is in parallel to the NT church, i.e. the New Covenant kingdom.
Almost all other lenses through which to examine the former kingdom--including it's split into two--end up a serious distraction to the truth of God. We are not going to learn much useful or beneficial for applying the text to secular politics (for example), by assuming that because the Old Covenant church was also a state, therefore states of all ages find in ancient Israel the model of ideal public order and governance.
All the uniqueness of God's covenant relation to this one nation of all in the whole earth is quite discounted by presuming the just application of his covenant monarchy to any earthly sovereignty or organization. God chooses his consort and offers her a covenant-relation; no nation or other group chooses for itself Israel's God to be its loving head; the Gibeonites (Jos.9) give us a kind of insight into what sort of relationship might come about if a nation offered itself to Israel's God: they became hewers of wood and fetchers of water, equivalent of slaves in other words. The nations qua nations are destined to be the Lord's footstool, his humiliated enemies.
Of course, it is still true that the nations surrounding the church today--as much as in old time, Dt.4:6-8 --might gain much from observing the true church carrying on and conducting its business in full compliance with Christ, who though he is King of all kings and Lord of all lords, is specially the Head of his mediatorial kingdom: the church. For example, the church contains a court (or a system of courts) where Holy Scripture obliges "due process;"
It sure is good if the secular government under which one lives has such a court, but the world's system does not recognize God speaking through Holy Scripture. The world gives selective due-process (an oxymoron of course) or no such thing; perhaps imperial Rome's example is a clear demonstration of the former. For all its vaunted world-wide justice, the Empire maintained very much the two-tiered justice system. However, partiality is forbidden the church today, and it was also forbidden to the church in the wilderness.
If the true church abandons due process (becoming less true thereby), why should it be surprising if the secular nation within which such a church has its dwelling becomes less fair and just? Has not the salt then lost its saltiness? Its influence is worthless, but the church is rather being influenced and becoming worldly. The northern tribes, which took the name Israel for itself and left Judah, and David's son, and the true altar of Israel--these tribes became a false church. They were Oholah (Ezk.23:4), and played the harlot against her husband. Before long, they were indistinguishable from all the other nations, bringing their gods within her borders after making her own golden calf anew.
But for a while, this false church was comparable to the rump of the true church in the southern kingdom. For a while, some pitiful witness was carried on in those borders, and there were notable efforts at reformation and reclamation. The ministries of Elijah and Elisha stand out in this respect, as they carried forth a witness against the lies and degeneracy, and called the faithful remnant away from contentment with the tragedy of apostate Israel. These faithful men (and others) conducted themselves in the Spirit even as far as Israel's king's confidences--with limited effect. But there continued to be a remnant of faith in Israel until the conclusion of that kingdom, as Hezekiah's call for a Passover of unity demonstrated, 2Chr.30, when an appreciable number of men made their appearance in Jerusalem. It was a last plea to the north for religious reconciliation and reformation, though it was received by many with scorn, v10; see 2Chr.31:1 for some visible effects of reformation carried out even in the north after this feast.
Hezekiah's great-grandson, Josiah, had to conduct his own reformation for the true church of Judah, see 2Chr.34:3. He made foray into the northern lands, as far as Naphtali, v6, but the work in the north was largely symbolic as there was no more church government there at all, not even a false one after captivity came. What remnant remained, v9, was vanishingly small. Josiah made his own effort at a Passover of unity, 2Chr.35, a celebration to rival his great-grandfather's v18; and while the north was represented, it seems the number of them that came from that territory was little reckoned (if comparing with the earlier report).
Yet, after Josiah the south was turned into Oholibah (Ezk.23:11ff), sister of Oholah and if possible even more determined to outdo her older sister inviting abomination into her bosom. The judgment promised by their Husband was terrible. Without a full recognition of the scope of divine promises, including the will of God to preserve and even resurrect a remnant (Ezk.36 & 37) it would seem that the true church wholly vanished from the earth. But God is faithful to his covenant, “Therefore say to the house of Israel, ‘Thus says the Lord God: “I do not do this for your sake, O house of Israel, but for My holy name’s sake," Ezk.36:22. There will always be a true church in the earth, made up of those who "humble themselves to come to Jerusalem." 2Chr.30:11.
Semper reformanda.
