We had a similar problem when we had film put on our lanai glass. The sky is reflected and they think it's open space and bam, they fly into the window. Cardinals are especially aggressive when they have a nest nearby. My neighbor found decals on line
Buy Window Decals, Butterfly Decal And Maple Leaf Decal At Windowalert which she placed on the windows and it worked (see interesting reason why it works). I had some of the film left over and cut it into long strips and hung it from the gutters. It wasn't unsightly and it did the job. Birds are wary of anything waving in their field of vision (hawks) and will avoid the window.