The gutter runs over, only at the downspout. so the downspout must be
clogged. I put a garden hose up the downspout but the second floor of
the house overhangs the first floor by 18" so there is a double turn in
the downspout, and I could only get about 10 feet of hose in.

Still I decided to see what the water would, and water came spraying out
from many places at the double turn. It shows that the connections
are only "waterproof" when the water is going down, not up.

Any idea how to push the hose beyond these two 100 or 110^ bends? I
put a sweeper nozzle on it just to help it around the corners, but it
didn't help.