IME, removing English ivy is a time consuming process. If you have done a good, thorough job initially - specially as far as root removal is concerned - you should be well on your way to having an ivy-free bed in spring. However, even small rooted chunks can develop into new plants, so you will need to be diligent in continuing to remove any signs of sprouts.

If you just removed the top growth without attempting to remove as much of the root system as you could possibly manage, you will continue to have problems.

25 years ago, I used Roundup. It took 3 applications a week or two apart or more to get rid of my ivy. It was thinner each time but that stuff is really hard to kill or remove.

I doubt that cardboard will do anything to get it out of there.