How should I fill in a missing strip of drywall that I uncovered after removing a piece of trim?
If that's actually subfloor (softwood t&amp;g) and finished floor we see in the photo, you really need to strip back some of the remaining drywall and mount a patch that spans the gap to solid framing. There's almost certainly a wall plate on top of the hardwood (odd, but explainable by past renovations), and a floor joist below. 

I'd cut 3/4" from both the top and bottom of your channel to find those, and then install a strip of drywall to them. Tape both joints in a nice wide, tapered manner and you'll have a good result.