Catalyzes the ATP-dependent phosphorylation of fructoselysine to fructoselysine 6-phosphate. May function in a fructoselysine degradation pathway that allows S.flexneri to grow on fructoselysine or psicoselysine. ATP + N(6)-(D-fructosyl)-L-lysine = ADP + H(+) + N(6)-(6-phospho-D-fructosyl)-L-lysine Carbohydrate metabolism; fructoselysine degradation; D-glucose 6-phosphate and lysine from fructoselysine: step 1/2. Monomer. Belongs to the carbohydrate kinase PfkB family.