So it looks like you can't even get to the actual duct. What I would do:
That "wood tunnel" between the end of the duct and the space above the bathroom ceiling can function as a duct. Cover end of the wood tunnel with a piece of drywall (or wood) with a hole in it that will accommodate a short piece of flex duct. Run that flex duct to your vent grill (technically called an air register). You will need to install a register boot in the ceiling behind the grill, it's what the duct and the air register attaches to:
