Second Kings picks right up at the end of First Kings. Israel in the north, and Judah in the south, continue in their decline. Judah as blessed with good kings here and there, delaying their dispersion to Babylon until 586BC. Israel had no good kings, and their speedy dispersion to Assyria occurred in 721BC. The non-writing prophets, Elijah & Elisha will be followed by the great writing prophets: Isaiah, Jeremiah, and others for whom Old Testament books are named.
The Book of Kings is one book in the Hebrew. Like Samuel, the Septuagint divided the book into two parts, and thus the 1st Kings and 2nd Kings we have in the English text today. The Hebrew title, Melakiym comes from the first word of the book: King (referring to David). The Septuagint renders our books of 1st & 2nd Samuel, 1st & 2nd Kings, as (Books of Kingdoms)
The text does not identify the author. The Talmud credits 1st & 2nd Kings to the prophet Jeremiah. There is good linguistic evidence for Jeremiah’s authorship when Kings is compared to Jeremiah & Lamentations. The phrases “to this day” found in 1st Kgs. 8:8 & 12:19 indicate an authorship prior to, or early in, the Babylonian captivity (586BC). The final paragraph (2nd Kgs. 25:27-30) is nearly identical to the ending of Jeremiah (Jer. 52:31-34), and appears to be an epilogue added by a later scribe.