Cats are unusual creatures as anyone that has one can attest. There is a phenomenon with feline diabetes where a cat might go into remission. It seems that their pancreas perks up and suddenly decides to work again. Often insulin treatment can be greatly reduced or even eliminated.
Another (auspicious) turn of events is when a cats diabetes can be corralled through change in diet. Often a cat fed a low carbohydrate-high protein diet will find its insulin so greatly reduced that they do not need insulin injections anymore.
The two scenarios are similar, except I believe that the first one applies more to Kitty when he went into remission. He was eating the same Science Diet Senior Dry cat food that he had eaten for years and yet his sugar numbers suddenly became normal without insulin. We did nothing special with his diet.
Many websites exist on proper foods to feed your cat. I found this good explanation of why to feed your cat wet, low carbohydrate food.
That being said, it is a big struggle to get Kitty to eat wet food, so we are just giving it occasionally. Ideally, I would like him to be on an all wet diet, but first and foremost is that he eat regular meals so that we can give him his insulin without interruption.