Sounds awful. Aside from vacuuming with an upholstery brush, which I'm sure you do anyway, I don't have any suggestions. They do sell special lamoshade sponges that are used dry, but I haven't tried them. I've included a link to one product.

However, if there's no hope of ever reducing the fly population (flypaper?), you might consider gradually moving to lamps that take non-fabric shades, such as glass, metal, mica, etc. Even without flies to worry about, fabric lampshades are generally fragile, and they're expensive to replace. Just a thought.

Here is a link that might be useful: Lampshade Cleaner Sponge

No offense but I can't imagine having flies in the house and not chasing them all down. I would have made sure the screens were all good and would have been yelling "close the d+++ door" all day. LOL

My husband's sister in law didn't worry about them. I hate flies and dreaded eating at their farm. For the most part the wife just thought they were part of living on a farm and nothing could be done about it.

I have used a Qtip dipped in pure bleach water on white polyester cloth lampshades. It's worked for me. Unless one has very expensive lampshades, most "silk" shades are polyester.

No don't use chlorine bleach even diluted in water on your lampshade. Its not rinsed out correct? So it remains eating away. Eventually the lampshade will just shred apart where the bleach touched it.