I think the style of curtains you have is the best option, but I would get some heavier ones (blackout or sun resistant) if you can afford them. Since it faces west, some of the new high-tech window film would be another option to reduce heat, glare and fading of your floors or furniture. With the crank style, it will be difficult to install any blinds or shades unless the crank folds into the mechanism to be less obvious. They are more contemporary windows, so I think cafe curtains are a little "off" and more old fashioned (although I've seen some really great ones.) I had corner west windows in a house but they were double paned. I just hung a single panel at each end for softness, pattern and color. Since they were stationary, I used a small bathroom towel rod to hold them. Looked great, but I have no pix to share.

Did you consider Roman shades? If custom is not in the budget there are reed ones that are inexpensive. If that is too bare looking use side hangs with them.

If hiding the “cheap” windows is a priority, I’d add another drapery panel (or 2) to the elbow of the L, so it appears you have two windows (versus stacking them on the left). I agree with using lined panels and you could also add sheers, for daytime privacy without blocking the light. If you use sheers, but still want to be able to close the drapes, you’ll need a double rod/two rods.

You could consider shades - there are styles that allow you to lower from the top or raise from the bottom. Graber blinds and Hunter Douglas both carry them. Another possibility- we have Hunter Douglas shades in a powder room that are fabric, but you can set them to open or shut sort of like blinds (The Silhouette line)

love the idea of Roman shades fr this space! Let us know what you decided to do. :)