What's the best way to repair this hardwood floor damage?
Remove the old, rotten wood and replace it with new stuff. This is a very common repair. 

I've never seen anybody use a jigsaw to remove hardwood strip flooring before but I'm sure it's been done. There's an Ask This Old House episode or two where they tackle this exact project, google it they're good videos, and Tom uses a drill with a 1" spade bit to remove as much material as possible and then uses a sharp chisel to break the piece of wood in half. This seems like good advice to me. 

Once you have one strip of wood removed then the others will be a little easier to remove because the tongue and grove are no longer locked. But some of your strip flooring pieces look very rotten and you might be able to just dig them out of there with a screw driver, chisel, etc. 

The last board to be installed will need to have it's groove edge chiseled into a rabbit so that it can drop into place. 

Good luck!

Also, if it was me, I wouldn't cut a big rectangular section of the flooring out as you described. I would only remove the pieces that are rotten and replace them with identical pieces of the same dimensions, length and species.