Window Tape

At my house we have a thing called a "fly stick" that is a sticky tube about a third of a meter long, that hangs from a plant hanger. Flies are supposed to land on it and get stuck. So far, after a month, there are none stuck. That's because the flies appear more interested in trying to fly through the window glass.

AHA!

Get some Transparent Double-Sided Sticky Tape for Windows, and attach strips near the edges of the windows, to block the view as little as possible. Replace a strip whenever one or more flies have become stuck to it. The window-sticky side might even be minimally sticky (like one of those "sticky note" pieces of paper), to make replacing strips easy.

This stuff might also be applied to the interior of the window frame, to increase the sticky area without interfering with the view at all.