Why can't you place it to the right of the tall windows? I also don't see a door leading to the back yard.

Looks to me like these are your choices:

The rear elevation of your home is great looking and it seems adding a permanent extended roof over your new patio will mar the appearance.

I know this may seem cheesy, but think about a retractable shade. Otherwise, I would suggest you consult with an architect about how to accomplish your covered patio goal.

Consider building building your patio the width of and in front of the roof line on the LEFT with a "greenhouse/sunroom" (clear or semiclear) roof that is above your upstairs windows but beneath your roofline.

You could have a gentle slope to the roof to move rainwater away from the house but your patio roof would not need to be square or rectangle; instead, you could create the shape of five sides of an octagon. You could have a ridge in the center of the patio roof closest to the house -- a ridge parallel to the two sides of the patio roof that are at a right angle to the house -- sloping the patio roof to left and right of that ridge. Then you could use the end of the ridge farther from the house as a center point from which you could slope the roof in three directions forming the three other sides of the octagon shaped patio roof.

Found (sortof) a 5/8 + shape of an octagon roof (it at least shows a ridge and three slopes from the ridge) ... but I'd certainly want much more in the way of support than this one -- it would not only be sturdier but look better as well..

http://www.royalite-mfg.com/skylights/catalog/pyramid/

ideas:

My daughter had a retractable awning on her last house over the deck, which was a two story deck since the lawn at the back of her house sloped enough so that you could go out the basement door right onto the lawn. It worked perfectly since a pergola would not have looked right and I believe it's the same for your house.

love this look but only projects 12 feet I believe.

the truth about a west facing patio.... the sun follows you as it sets. you can put on a roof but the light will still be in the patio, just as you have a roof on your living room but the sun still comes into the windows. build on a different side of the house is sometimes an option, we went south for our deck and screen porch.. but looks like you have tight lot lines.

i'd go with building a freestanding pavilion in your large enough back yard. lemme look for photos.

something like this with its back to the west, facing your house. pretty patio separating them. far enough away so you don't lose the light in your living room.

this one does a roller shade on the back... an option for you... it would be open the rest of the time.

Old post but it came up on my feed.

What did you decide? Pics?

Dig Doug's Designs suggestion for a forward facing gable would have been my choice.