Catalyzes the phosphorylation of L-xylulose and 3-keto-L-gulonate. Is involved in L-lyxose utilization via xylulose, and may also be involved in the utilization of 2,3-diketo-L-gulonate. ATP + L-xylulose = ADP + H(+) + L-xylulose 5-phosphate 3-dehydro-L-gulonate + ATP = 3-dehydro-L-gulonate 6-phosphate + ADP + H(+) Optimum pH is 8.4 with L-xylulose as substrate. Homodimer. Belongs to the FGGY kinase family.