Looks like a no brainer to me. Put a shed roof over it with two posts outboard to support the roof. screw a 2 x to the T1-11 level, saw cut and remove. Flash under and over the new shingle roof. Screw a 2 x 8 ridge plate to the studs of the house with 2- 3/8 x 3.5 inch Lag screws per stud, and frame her out with 2x6 rafters. Put vents on the sides. I would make it bigger than the well by at least a foot for wind blown rain, and with the posts you could even install trellis for some privacy, and partial wind block.
Good Building.