Roll out your wire and carefully form a tall cylinder about 14.5" wide. Don't make it too wide as it won't fit your laundry bag.
Using wire cutters, carefully remove any excess wire, leaving about a 1.5" overlap. You may need to trim about 10" off the bottom of your cylinder as well, so the laundry bag sits inside the basket and barely brushes the base.
Using needle nose pliers, bend the cut metal around the other wire mesh so it's secured together. Make sure all the little prongs are bent so they don't poke you. Do the same with the mesh you cut at the bottom. Bend all the prongs outward so it sits evenly on the wood.
Now, place the wire cylinder in the middle of the wood base. If you don't want a natural base this is the time to stain it or paint it and let it dry.
Using hot glue, liberally pour hot glue along the edges of the wire, going along the entire cylinder so the wire mesh sits in hot glue. Let dry.
Now place a thin layer of hot glue along the base of the wire and add your rope all around, several times so all the glue is covered and your base looks complete.
If you want to add casters you can flip over the hamper and screw in your swivel casters. Insert laundry bag and use!