How are these two roof structures joined over the entrance? A gable with pitched roof running back--possible a sunroof--could unify the structure while creating a very interesting interior space. Creative attention to engineering should make this a manageable expense. Alternatively, frame the same gable idea as an extension of the pergola and create the illusion of a unified roof structure by masking the divide. Have fun!

Is the gap covered with a flat roof now or open to the air above?

This house appears to have been reworked a couple of times. The garage pergola is a giveaway. Probably started life as a ranch with detached 2-car garage, then got the mansard treatment. Then received the connecting fenced courtyard and pergola. Just guessing.

If you provide a photo showing the whole house, we can better comment.

You might be able to do a mansard roof, but that might be too heavy on a ranch. But here is one attractive example:

Replacing the current roof with a standard gable would look the best:

You might be able to get preformed trusses to reduce the cost.

can you take some photos from further back? ... from what I can see in these photos, I like the look of your home. I think it 'works'