Walls and Ceilings - Repairing drywall openeing around new door

I just replaced a window with the same width patio door. The corners of the drywall around the original window were bullnose which means that the edges of the drywall were scraped away extensively. So, now I have a 1"+ gap between the interior door frame (vinyl clad) and the edge of the drywall.

Should I tape over this, attaching the tape to the door frame and then repair it with mud or should I first add a narrow section of drywall along the edged of the door? I'm afraid that it will be difficult to neatly cut two 1" x 6'8" lengths. Also, should I use some sort of a spacer beside the frame to avoid "attaching" the repair to the door? There will be trim added later.

Thanks for your help.

Since you are covering it with trim I recommend filling it tightly wih insulation
or one of the many foam products. If you use foam get one with minimal expansion.
I am a remodeler and have done this more times than I can recall.