This afternoon, I completed what I think was the final step of the refurbishment.
Using a vinegar solution, I cleaned all of the calcium-lime stains off the plastic top of the cooler. This is only cosmetic, unlike some of the other encrustations that affected the performance or damaged the structure of the cooler. It does look much tidier now.
Now all we need is some hot weather and we'll be ready to go. Looking at the forecast, we won't have long to wait.