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Lay down two of the bookcases running parallel to each other 59 degree's apart. The third bookcase will be placed on the end of the first two bookcases to cover the entire 59 degree span. Using a hex key, remove the eight screws holding in the end pieces. Measure and mark 29-1/2" from the center to make the third bookcase 59" long.
Cut the ends off using a circular saw. Cut wood scraps to fill and cover the now hollow sides of the bookcase. Use wood glue, and then screw the blocks in. Pre-drill holes to remount the end pieces. Screw the end pieces in their new place using the hex screws previously removed.
Measure, cut and attach new 1/4" plywood backers to cover one side of each bookcase. Cut plywood sheets to cover the bookcase and to provide a surface for the mattress. Using a circular saw, cut one piece of 3/4"plywood to 48" x 60", and then lay it down across the end with the shortened bookcase. Overhang 1/2" on the sides and 1-1/2" on the end. Cut 3/4" plywood to 42-1/2" x 73-3/4" using a circular saw, and then lay it across the bookcases at the other end with the sides equally overhanging the edge. Measure and mark the length of the couch arms from the back end of the wider platform. Cut in about 6-1/8" on both sides, creating a "T" shape that matches the width of the other platform where the ends meet. Fill any imperfections with putty, and then prime all bookcase and plywood surfaces, including the inside of the backers. Paint all prime all surfaces.
Remove the sleeper sofa mattress, mechanics and wheels. Cut back the fabric with a box cutter, and then cut off the sofa kick plate using a circular saw. Using 3/4" plywood, construct two boxes to 34" x 5" x 8" to hold the couch above the new bed frame. Using large screws, joinery plates and construction adhesive, attach the boxes to the bottoms of each sofa arm. Cut two triangular stops from wood scraps. These will be nailed to the plywood base to stop the box spring from sliding under the sofa back headboard.
Using a turkey carver, cut two 4" thick foam inserts 22" x 21" that will fill in the excess space between the sofa arm and the mattress and box spring. Cover all couch surfaces with new fabric, including the newly attached lifter boxes and foam inserts. Use a staple gun to attach the ends of the fabric and hide them under upholstered cover plates on the arms. Make sure staples are only placed where they will not be seen, such as the back and underside of the couch.
Screw the plywood platform into the tops of the bookcases, holding it firmly in place. Screw up through the bottom of the platform to attach the sofa lifter boxes. Nail the stopper triangles in place under the sofa back headboard.