Catalyzes the ATP-dependent phosphorylation of the 3-deoxy-D-manno-octulosonic acid (Kdo) residue in Kdo-lipid IV(A) at the 4-OH position. To a lesser extent, can use GTP instead of ATP as substrate. alpha-Kdo-(2->6)-lipid IVA + ATP = 4-O-phospho-alpha-Kdo-(2->6)-lipid IVA + ADP + H(+) Optimum pH is 7.5. Bacterial outer membrane biogenesis; LPS core biosynthesis. Belongs to the protein kinase superfamily. KdkA/RfaP family.