If it were me I'd go under that sidewalk to the other planter as well to keep staining off of the driveway.
But if you don't like digging, yeah just connect to the down spout on the right and T it at the next one then wrap it around the corner ending at driveway. Keep it tight to house and plants will screen it.
I agree too small a planter for the asian jasmine. Just get some nice shade 5gal plants to fill the space.