Here are some canisters already cut, one with batting already glued to it. The batting should extend from the top edge to the bottom. Do not wrap the batting around the top edge. The lid will fit better if you don't add extra thickness. I used some poly batting that I had around. I usually use cotton, just because I like cotton. But I think that is up to you. I used scraps that I had available. In fact I think if you use different fabric scraps for each part, it would make a very cute sewing kit! Since I am relatively new to quilting I don't have very many large scraps yet, but I am working on it! LOL
Use a glue stick and rub it all over the outside of the container. Don't overlap your batting but try to make it abut evenly. If you find that you cut the batting a bit short, you can stitch it together with a lacing stitch, like shoestrings in eyelets.
The dimensions of my batting are 12 3/4" x 3 3/4". If you cut the canister a different height or use the large size, you will have to adjust your measurements accordingly.