How do I repair loose ductwork?
If you don't mind a little extra thickness, what about hanging some L brackets over the edge of your finish flooring, and then screwing the ducts to the L? Then, you can put the duct cover back on right over the lip of the L, covering it up. 

Bad ASCII art time!

   --+       +--  L brackets
==== ||    || ====Floor
====  |    |  ====Subfloor
       \  /  
      |DUCT


I have no idea how you'd hold the duct in place while screwing it, mind you, unless you can get access from below. Maybe you could glue a rope to the inside of the duct and have someone yank it back up into place.