Drill a 2-inch hole through the center of one of the 5-by-5-inch sheets. The center of the hole is 2 1/2 inches away from each 5-inch edge. This is for the door on your cage.
Set two 5-by-5-inch pieces of melamine on edge, parallel and 8 inches apart. Lay the 8-inch melamine between them so the edges are flush. Drill two 3/32-inch pilot holes through each 5-inch piece and into the 5-inch ends of the 8-inch piece. Drill a 3/8-inch-deep counter-sink hole in each of the pilot holes to hide the screw heads and screw the three pieces together.
Screw the plywood to the melamine so it covers the 2-inch hole. The screw is 1/2 inch above the top of the hole. Use the 1-inch screw. This is the door to your cage. It will pivot around the screw to cover the hole and to uncover the hole providing access to the cage.
Lay the screen mesh over the edges of the 5-by-5-inch melamine sheets so the edges are flush. You are covering the three open sides so you will have to fold the mesh over the two pairs of corners. Staple the mesh to the melamine using one staple every inch. Leave no opening into the cage with the edges of the mesh.