They could possibly live alongside hens, given enough space, but in your situation it would add to your problems, Dye. You've posted about the mud in the run, and since you have to tip out the ducks filthy water every day or so, this will make the ground into a swamp. They are much messier than hens in any case, and will add to your rat problems. They will need a coop of their own, at ground level, as they can't roost and perch with your hens. They will need herding into the coop every evening at dusk as they don't take themselves off to bed like chickens do. They eat soaked wheat, or special duck pellets, not layers pellets, and in a mixed run the chickens will eat it too, and the lower protein and calcium content will unbalance their diet and egg production will go down.
I understand your kind impulse to save these ducks, but perhaps your friend is piling on the pressure for you to solve what is really his problem, not yours? Unless you can provide them with a large enough rat proof run of their own, my advice would be to think it through before rushing in to taking them.