Use a stud finder to locate two studs in the upstairs hallway nearest the laundry-area side of the house. Mark the center of each stud with a pencil.
Remove the baseboard from the wall below the area where the laundry chute will be installed. It can be pried away from the wall, with little or no damage, using a flathead screwdriver.
Measure to a height of 42 inches from the floor. Cut the drywall with a utility knife down the center of each stud. Cut across the top and bottom of the drywall and remove the 42-inch piece.
Cut out the baseboard that was exposed by removing the drywall with a reciprocating saw.
Cut a hole through the floor beneath the missing baseboard so that the duct can slide through.
Install the cut-out baseboard at the top of the 42-inch cut-out space. Secure it to the wall studs with wood screws angled upward at 45 degrees.
Connect the 3-1/4-inch-by-12-inch (or 3-1/4-inch-by-14-inch) galvanized heating duct to the 90-degree elbow and expand the register opening from 6 inches to the size of your clothes chute's door. Use tin snips to make the cuts.
Slide the duct into place and secure it with sheet-metal screws. The screws should be installed into the repurposed baseboard and the wall studs.
Cover any sharp edges and screwheads, as well as any joints, with duct tape to prevent clothes from getting torn or stuck.
Cut the drywall piece you removed earlier so it will fit the opening in the wall. Use drywall tape to hold it in place. Cover it with a thin layer of drywall compound and wait for it to dry.
Sand the drywall compound smooth; then prime and paint the wall to match the original color.
Screw the clothes chute's door to the wall studs.