Use a lint roller to clean screen doors
If your house has a screen door, you know better than anyone how impossible they are to clean. Every little speck of dust and spider web clings to it, and wiping it down with a rag makes absolutely no difference. The little holes make it impossible to remove all the debris with any sort of efficiency. Nonetheless, here is a trick that may actually help.
You can try using a lint roller to get all the dirt off. If you do this first, then wiping it down with some cleaner and a wet cloth will be much more effective. Could this be the secret to finally having clean screen doors? You will have to try it out and see. Otherwise, a power washer might be your only alternative.