While there's no non-biblical "proof" for the United Monarchy, there's also no reason why we can't cite the Hebrew Bible/Old Testament as evidence. To avoid a lengthy discussion about the compositional history of the Book of Kings, I will simply point out that scholars generally accept the authenticity of Solomon's administrative lists in 1 Kings 4; the second list (1 Kings 4:7-19), in particular, names the officials whom Solomon had appointed over the territories of Israel. The lists themselves are consistent with the structure and language of other administrative documents from the ancient Near East, while the inclusion of old Canaanite institutions and non-Yahwistic names strongly suggest an early provenance. In addition, the borders of the districts seem to be reflected in the archaeological distribution of the famous tripartite pillar buildings.

Sources and further reading: Joseph Naveh, "Nameless People," Israel Exploration Journal 40, no. 2/3 (1990): 108-23; Richard S. Hess, "The Form and Structure of the Solomonic District List in 1 Kings 4:7-19," in Crossing Boundaries and Linking Horizons: Studies in Honor of Michael C. Astour on His 80th Birthday, ed. Gordon D. Young et al. (Bethesda: CDL Press, 1997), 279-91; Baruch Halpern, David's Secret Demons: Messiah, Murderer, Traitor, King (Grand Rapids and Cambridge: William B. Eerdmans, 2001), 413-7; Jeffrey R. Blakely, "Reconciling Two Maps: Archaeological Evidence for the Kingdoms of David and Solomon," Bulletin of the American Schools of Oriental Research 327 (2002): 49-54; Lawrence E. Stager, "The Patrimonial Kingdom of Solomon," in Symbiosis, Symbolism, and the Power of the Past: Canaan, Ancient Israel, and Their Neighbors from the Late Bronze Age through Roman Palaestina, ed. William G. Dever and Seymour Gitin (Winona Lake: Eisenbrauns, 2003), 63-74; William M. Schniedewind, How the Bible Became a Book: The Textualization of Ancient Israel (Cambridge: Cambridge University Press, 2004), 59f.; David M. Carr, The Formation of the Hebrew Bible: A New Reconstruction (Oxford and New York: Oxford University Press, 2011), 369.

Why can't we say that the Bible can be taken as proof? Legitimate question as this isn't my field of study.

As far as I know, from my old days of studying the Bible loosely, many of the older Biblical works (Joshua, Judges, Samuel, Kings, Isaiah, etc.) which speak of the Kingdom of Israel and Judea are dated around 800 BCE. Obviously we must ignore the supernatural elements but would the more mundane details be really in that much question? I'm no ancient historian, hence me asking, but is it really worth throwing out hundreds of pages of firsthand primary material from the 8th century BCE because of some unreliable elements that are easy to spot? If anything it seems like a valuable primary source into the lives of people in that region at that time when we ignore the whole pillars of fire nonsense.