Cardboard isn't the best; it insulates well, so heat becomes a problem. If your closet is only 20'', then you can make do with what you've got. Are you UK or US, that would help me figure out where you can get materials and what to do with them.
Sure, I'll help if I can. 20'' seems a bit narrow, I'd go at least 24''. Obviously there's details missing, but I'll help you with what I can. If you want to talk live, my number is 8284301843 or 7044720166.