She can be cleaned up with a quick shallow soak in warm soapy water, but be sure to towel dry her and let her dry inside completely. I would imagine her messy bottom is due to her illness. Make sure that she is eating and drinking, offer some chick vitamins to her water, and see if she will eat some chopped egg. How old is your hen? With lameness and a droopy eye, Mareks comes to mind, but so many of the reproductive disorders (internal laying, egg yolk peritonitis, ascites) can cause sleepiness, lameness, poor appetite, weight loss in the chest, and runny droppings. Let us know how she is doing later.