Residential Remodel

Stair landing on main floor is spongy. removed part of basement ceiling to view. Discovered rim joist was cut for duct work to run through. Its currently fastened with 2 by 10 hangers and the nails are not standard nails. The total of joists are 12 and all need proper joists nails. The working room is 6'6" x 15'. All joists are 2 x10"x 8'. They require brace blocking in caviting for racking and extra nails for the sister overhangs. Also i am going to use screw jacks with a 4 by 4 posts for support to level floor and reinstall joist nails in all hangers. For the hangers i can't reach by the duct work im planning on using ledger lock screws on the back side of rim joist to fasten to the 2 by 10 joist. I also will add additional 2 by 10's for support of ceiling along existing rim joist and fasten properly with nails, and ledger lock screws as needed. Re-install 5/8 fire rated drywall, tape, texture to finish.