Kylose kinase that mediates the 2-O-phosphorylation of xylose in the glycosaminoglycan-protein linkage region of proteoglycans. 3-O-(beta-D-galactosyl-(1->3)-beta-D-galactosyl-(1->4)-beta-D-xylosyl)-L-seryl-[protein] + ATP = 3-O-(beta-D-galactosyl-(1->3)-beta-D-galactosyl-(1->4)-beta-D-2-O-phosphoxylosyl)-L-seryl-[protein] + ADP + H(+) Belongs to the FAM20 family.