Catalyzes the phosphorylation of heptose(I) of the outer membrane lipopolysaccharide core. The phosphorylation of the lipopolysaccharide core seems to occur prior to translocation to the periplasm and attachment of O-antigen. Also has protein-tyrosine kinase activity: autophosphorylates on all Tyr residues; in vitro can phosphorylate poly(Glu,Tyr). ATP + L-tyrosyl-[protein] = ADP + H(+) + O-phospho-L-tyrosyl-[protein] For lipopolysaccharide kinase activity. Bacterial outer membrane biogenesis; LPS core biosynthesis. Phosphorylated protein. Lack of viability. Shares similarity with eukaryotic protein kinases in the conserved tyrosine kinase functional motifs. Belongs to the protein kinase superfamily. KdkA/RfaP family.