17 x 6? I'd do the whole thing as a combined open air coop and run.
You've space for maybe 10 birds there, I'd keep it to 8. Y
our heaviest rainfall are June, July, August, falling off in September. Winds during that period are primarily out of the south, south east, or south west. Hopefully, your dog kennel runs North/South. If so, I'd throw up a wall on the south side, plus a short portion of the east and west sides to form an enclosed "c" shape.
Whether you want to do a raised coop (and thus, only half walls) or run the walls all the way to the ground is up to you. There are tradeoffs to either design. Unfortunately, you are going to have to walk into the coop to gather eggs - chain link isn't the easiest material to cut good holes in w/o specialized tools and extra materials. For that reason, I'd lean towards a raised coop, to get things up to a comfortable height for you to gather from.
Material is also up to you - the chain fence provides much of the structure, but isn't exactly conducive to attachment via nails or screws. I'd have to stare at the hardware aisle for a while to have a recommend there - and how I'd attach a polycarbonate panel is obviously different than how I might attach plywood or even sheet metal - but if you du use metal or polycarb, 2 sheets at 2' (nominal width) and 12' length, two cuts later, and you've got 3' for the left side, 6' across the back, and 3' for the right side - which encloses plenty of space for 8 birds. Hang two or three nest boxes (plastic "milk crates" might be good for this), a perch rod or two above that, and you've got a very practical hen house for all but hurricane conditions.
Then using 2x8s, 24" oc and secured to the top of the kennel by
two hole straps (size these based on the diameter of the pipe at the top of your run) and probably 6' of roofing in width, leaving the rest open. Span is short enough, you could lay them on the flats and pretend they are purlins. Will sag in time, but not much.
You can do that with either 2 pieces of 39" x 8' roof steel, or 3 pieces of 26" x 8' roof steel. Whether or not to cover the rest is budget and predators. Optionally, cover about 7' of the cage by running the steel parallel with the long axis of the cage, leaving about a 1' overhang in three dimensions.
Vaguely (yes, I used a mickey mouse paint program instead of doing it on CAD. I'm being a lazy ass tonight)