Drafty gap found in drywall corner during bathroom remodel
Assuming you have little to no batt insulation at that corner (and there's nothing snug that would hold-up a strip of the same if stuffed into the crack with a putty knife): 

Get a 4"-8" wide strip of metal screen (as used in a screen door) and fold it in half.  Insert it into the crack.  Tape it open, to the left and right sides of the gap, respectively, thus creating something onto which an insulating spray foam will attach without falling to the floor of the stud bay. Spray foam using the straw-like attachment, let cure, and trim any protruding excess and in such a way that one forms a proper inside corner for the drywall tape. Note: this stuff expands greatly so go easy with it.

Then, using self-adhesive fiberglass drywall mesh, repair as one would normally do for a small hole.