No, you cannot make cheap and healthy cat food at home.
Because cats are carnivores. The natural and healthiest cat diest is purely composed of meat, skin, bones and other animal byproducts. Meat is expensive. That's why all cheap cat foods contain a lot of grain and only a little meat, because grain is much cheaper. This grain is exactly what made your cat ill in the first place.
The most common stones cats are prone to developing are called struvite stones and caused by a chemical imbalance in the blood of the cat due to unhealthy food and too litle water intake. Please be aware that there are other types of kidney stones and you should consult your vet when changing the diet or treatment of your cat. Kidney stones are very painful and cats with such stones have a tendency to pee everywhere (not to annoy you but as their call for help) and cost a lot at the vet. Unfortunately, due to people's unwillingness to pay for good cat food, more and more cats develop those stones.
Richard Goldstein, DVM, associate professor of small animal medicine at Cornell University’s College of Veterinary Medicine states in this article:
we’ve observed that [the formation of struvite stones] tends to occur frequently in domesticated cats, especially in those that are not very active, don’t take in enough fluids, and don’t urinate enough.
So one precaution against the formation of new stones can be soaking your cats food in a bit of water to increase the overall water intake.
The most natural diet (that also prevents the development of kidney stones) would be feeding your cat fresh, whole rats, mice, chicks and other small animals. I don't see how this could possibly be cheaper than buying good quality cat food.
Many people get the idea of feeding their cats cooked chicken or fish. That's actually unhealthy, especially if you seasoned it with salt. Cats should not eat any salt or other seasoning, especially since consuming salt increases the risk of kidney stones. This would also be a very monotonous diet that causes malnutrition.
The only feasable solution is looking for a different brand of cat food that is designed to prevent urinary stones. Since there are different types of urinary stones, you should consult your vet about the change in diet and schedule a routine appointment to check if the new food actually works.
This routine checkup is important because most cheaper food brands don't undergo extensive testing of their actual performance in reducing urinary stones, as this article states:
Diets at grocery or pet stores marketed for urinary tract health in cats focus only on some factors that reduce the risk of struvite stones in cats, but these diets do not undergo testing for activity product or RSS ["relative supersaturation" - the measurement of how likely a stone is to form in urine]. They may be a compromise for cats with a history of struvite crystals or stones that won’t eat a therapeutic diet or if therapeutic diets are too expensive. However, the down side is that they are likely not as effective. These diets will not be helpful for other stone types and may increase the risk of other stone types in predisposed cats.
So don't hold your hopes of finding cheap and healthy cat food very high. Cheap food is the equivalent of junk food for cats. If you want healthy food, you need to pay for the higher quality. In the end it's still cheaper than the vet bills cheap food would cause. Also keep in mind how long such a big pack of food lasts you. Broken down to weekly costs it isn't that much anymore.