Measure the frame of your sofa bed under the mattress. You will likely need to pull up the mattress to do this. You may wish to leave the mattress to one side until you are done with your repair.
Cut the amount of plywood that you would need to cover your sofa bed frame.
Cut the plywood into three pieces that will sit horizontally in your sofa bed frame. When you are done, you should have three pieces of plywood that will sit comfortably inside your sofa bed or underneath it when it is folded up.
Drill three holes along one edge of two of the plywood segments. These holes should be spaced evenly.
Drill three corresponding holes along both edges of the third plywood segment.
Lay the three pieces of wood on the floor in the configuration that they will take when they rest on the sofa bed frame.
Lace the wire between each set of holes. You are going to be creating links that allow the pieces of plywood to fan fold together for easy storage. Remember to leave enough slack in the links so that the pieces will fold over neatly. It make take a little bit of work to get the tension right.
Trim any protruding wire ends with the wire cutters.
Lay the finished product out flat on the sofa bed frame.
Put the mattress on top of the new, reinforced frame.