Dating the foundation of the King's House is not straightforward. In the earliest days of Gondor, the realm was ruled jointly by the brothers Isildur and Anárion. Each of these had their own separate dwelling, with Isildur in Minas Ithil and Anárion in Minas Anor. It was not until after the War of the Last Alliance that Gondor had a unified Kingship under Anárion's son Meneldil. He ruled from III 2, and it was presumably during his time that the King's House was established in the royal city of Osgiliath. The most we can say definitively is that the House was in Osgiliath when Ostoher came to the throne four centuries later (because we have a record of Ostoher changing this arrangement).