You need a car brake reliner, tubby. Most of them send the jobs to a city or regional specialist these days, but some still do it themselves - I don't think it matters very much whether it is done locally or not. To get the job done right the first time, you should figure out how thick the lining material should be. That means adjusting the band adjuster in the loosening direction, springing the band to make its diameter as big as the linkage will allow, and measuring the internal diameter of the band. Then measure the outside diameter of the brake drum inside the band. Give both dimensions to the reliner, and he/she can figure out how thick the lining can be.
One thing you might try, is talking to Cox or a local rideon dealer, and see if there is somewhere they usually send these jobs. It might save you the hassle of accidentally choosing an incompetent supplier. That mess you currently have, is the result of someone either trying to do it himself, or choosing a really awful relining shop. The lining material needs to be solid material, bonded to the band, and I think that was woven material, probably glued (badly) to the band. As you said, it might even have been a piece of woolen carpet.
