Remove the door and basket inside the linen chute cabinet. Put the cabinet roughly in place. Clamp it to the existing cabinet. Check the cabinet is level and plumb. You may need to adjust the feet to make it level.
Screw the laundry chute cabinet to the wall at the marks you made to indicate stud placements. Pre-drill a small hole through one cabinet into the other. Screw cabinets together using a screw that is just long enough to attach the 2 cabinets but short enough not stick out the other side. Re-attach the door and basket.