Half inch metal hardware cloth seems to keep them out.
My first thought is, set aside some time, pull bedding out, look for how they are getting in and repair that. You may need to build a door or something with the hardware cloth to block their entrance.
Then replace the bedding.
Set up some water traps (a bucket half filled with water, with a dowel threaded through two level holes in the top, with peanut butter on the dowel) near the coop. The idea is the rat is attracted to the peanut butter, the dowel flips it into the water and it drowns. Or if that sounds unkind, if you know someone who is a good shot and can kill them, but then you must account for any ammunition so the ducks won't eat it and get hardware disease (metal toxicity, which is often fatal to the duck).
There is a Predators and Pests forum, too, you may be able to get more help there.
There are also smoke bombs you can place in their holes - but domestic animals are not to be near the area for four hours, and they are not to be used near structures.