Involved in the catabolism of L-rhamnose (6-deoxy-L-mannose). Catalyzes the transfer of the gamma-phosphate group from ATP to the 1-hydroxyl group of L-rhamnulose to yield L-rhamnulose 1-phosphate. ATP + L-rhamnulose = ADP + H(+) + L-rhamnulose 1-phosphate Carbohydrate degradation; L-rhamnose degradation; glycerone phosphate from L-rhamnose: step 2/3. Belongs to the rhamnulokinase family.