There is, of course, another account of this history, the book of Chronicles, 1 & 2 Chronicles/Paralipomena - the Greek name means the things left out. however, the problem with Chronicles is that not only does it include material not found in Samuel or Kings but the portrait of David and Solomon found in Chronicles is markedly different to that in Samuel and Kings. Indeed, Chronicles is mostly concerned with the Temple cult and so David here appears in a very idealised almost priestly form and Solomon after him. It omits the story of Bathsheba, Absalom's rebellion and other less than flattering accounts of David. So Chronicles account of these stories is generally not regarded as having much historical merit at all.
Related to Chronicles is Ezra-Nehemiah, which in Jewish bibles is counted one book but in Christian bibles is counted as two. And just like Chronicles, Ezra-Nehemiah is very problematic for doing any history. Parts of both texts are related in first person, however neither Ezra nor Nehemiah seem to be aware of each other's existence, which is odd to say the least because they appear to be contemporaries in the text. Is one fiction and the other historical? The problem, which is which? And what then is the relationship of Ezra-Nehemiah to Haggai and Zechariah which appear to have another account of the rebuilding of Jerusalem and the Temple. Is there history in any of these four texts? And then there is 1 Esdras which is another version of the material in Ezra-Nehemiah and some from Chronicles with some original content of its own. In other words, it's mostly derivative and not without problems at that.
Whatever one can say about Esther, one certainty is that this is a fiction designed to provide a scriptural warrant for Purim. Tobit, any history there, most unlikely. Judith? If that's history I'll eat it. Ruth is a lovely story but can it really serves as a history especially as it concerns the ancestors of David, who is likely to be a fictional character. Job? To read Job as history is a meaningless exercise. The text is not interested in providing historical details such as when or even where Job lived. The text, of course, is a script and Job is likely to be a sacred drama of some sort. Jubilees is a retelling of much of Torah.
When we turn to the rest of the biblical literature it gives little or no help in writing a history of events. Psalms, Proverbs, Qohelet, Wisdom and Sirach might be useful for writing a history of religion. Lamentations and Song of Songs are of little use to history work. Daniel is prophetic commentary on the events around Antiochus Epiphanes in the 2nd century BCE and so useful in that sense. 4 Ezra is prophetic commentary on the destruction of the Second Temple.
Only 1 and 2 Maccabees seem to be doing any real sort of history and even there we have to be careful. 2 Maccabees is more interested in a theological interpretation of events and does not cover the same breadth as 1 Maccabees. 3 and 4 Maccabees on the other hand are not interested in history, as we understand it, at all.
So if we're wanting to write an Israelite history, most of the Old Testament literature is of little use for getting details of key events and persons.