Open the sofa bed, and use a tape measure to determine the length and width of the frame under the mattress. Using a circular saw, cut three 1/4-inch plywood pieces to match the measurements of the frame.
Drill three holes, evenly spaced, across one horizontal edge of two of your plywood pieces. Drill three holes on both horizontal edges of your third plywood piece. Space the holes on the third piece so that they correspond with the previously drilled holes in the first two plywood segments.
Position the three plywood pieces on a work surface or on the floor. Position the plywood piece with the holes drilled into both ends in the middle. Put the drilled edges against each other in the same configuration as they would lie on the sofa bed frame.
Run wire through each set of holes. Lace the wiring to link the plywood pieces together, and then tie off the wire. Allow enough slack when lacing the wires to enable the pieces to fold over.
Use wire cutters to trim any excess wiring. Lay the wood pieces flat on top of your sofa bed frame. Position the mattress on top of the pieces and sofa frame.