Not only does God compare Judah to a disobedient wife, he also compares it to a filthy loincloth that is now unusable, though it used to be his (God's) loincloth. That's right—God says, "For as the loincloth clings to one's loins, so I made the whole house of Israel and the whole house of Judah cling to me, says the Lord, in order that they might be for me a people, a name, a praise, and a glory. But they would not listen" (13:11).
So God's suggesting that he and Judah should really be on pretty intimate terms. Judah would be, in the modern world, like God's pants (though a loincloth is a little more personal than pants, since it's effectively like shorts and underwear rolled into one). But they haven't been good pants. Judah's not a freshly pressed pair of Dockers khakis. Judah's a horribly ruined pair of Dockers khakis. So God will cast it away for its sins into the giant dumpster of Babylon.
But maybe Babylon is actually more of a Laundromat in this case, since Judah will eventually be restored, and peace will reign. God makes his break with Judah sound permanent at times, but he says it really isn't.