Set two 3/4-by-36-by-48-inch plywood boards on top of each other. Screw the boards together by drilling 1 1/4-inch wood screws into the boards in a 12-inch grid.
Arrange six 2-by-2-by-12-inch wood posts evenly underneath the plywood boards. Drill two holes down through the plywood boards and into each of the posts. Attach the boards to the posts with 4-inch wood screws.
Set two 3/4-by-24-by-48-inch plywood boards on top of each other. Attach them together in the same manner as the other plywood boards.
Position the newly attached plywood boards behind the couch frame, perpendicular to the couch frame. Hold the vertical plywood board, which is the backrest, so that the bottom of the board is even with the bottom of the seat board.
Set six 3-inch metal L-brackets against the rear of the backrest board and the underside of the seat board. Attach the metal L-brackets to the plywood with 1 1/4-inch wood screws. Set six more 3-inch L-brackets evenly distributed on the top of the platforms between the top of the seatboard and the front of the backrest. Attach the brackets in place with 1 1/4-inch wood screws.