At least some of the issue seemed to be with the screens. I think that the dirt in the air first goes thru the screens and some of it stays on the screens. To help clean the screens, I used a bottle of Windex Outdoor Glass Cleaner. This product attaches to the hose. First you rinse the window with clear water, then add the soap solution, then rinse it off with clear water again. Since this is done from the outside (with the windows all closed), the screens get washed off too.
It took one bottle and 1.5 hours to wash all the windows this way.
Then I washed each of the windows, outside first, then inside, using a bottle of Windex (or a generic equivalent) and a squeegee. This took another 5 hours, over two days, for all the windows.
The hardest window to clean was the second story fixed window in the computer room.
This window seems to have water spots on the outside that will not wash off. Maybe the result of hard water or acid rain.
