Why not extend the downspout exit out further from the foundation?
You could bury a large pipe at the base to catch the water, then transfer it out to a street, or away from any other buildings.
I think that the water from the higher one rushing into the lower one will just overflow all over the place, and still be flooding too close to your home.
I have seen gutters added to buildings similar to yours, with a slight slope from a higher one to a lower one, with a much larger lower gutter to be able to hold the extra water.
ED
You could bury a large pipe at the base to catch the water, then transfer it out to a street, or away from any other buildings.
I think that the water from the higher one rushing into the lower one will just overflow all over the place, and still be flooding too close to your home.
I have seen gutters added to buildings similar to yours, with a slight slope from a higher one to a lower one, with a much larger lower gutter to be able to hold the extra water.
ED