It can be environmental allergy to many things in the home.And it could even be environmental allergy to washing powder...not sure what to change to use for that though.
If this were my cat I would remove all chemicals from the house entirely. No hair spray no chemical cleaners at all, nothing with any scent to it, and so on. I would find a place to store the laundry detergent so that there's no chance it will be in the air, and switch to a liquid form that will not put powder into the air. No cleansers at all -- just use white vinegar and water to clean. Cheaper and just as effective for almost everything. Dish soap- make sure it doesn't have any fragrance in it. Same with human soap. Go as simple as possible.
Keep in mind that only feeding chicken or turkey is not a balanced diet (if you mean just the meat, rather than a chicken or turkey based canned food). If you want to do an elimination diet you need to go down to only one, not two at a time.
Keep in mind also that if your cat has kidney issues, feeding a food that is as low as possible in phosphorus is very important. You may have to do some research to find that food, but that is one thing you need to focus on.
It's hard, having allergy issues and other issues at the same time. You may have to sacrifice what is best for one in order to prioritize the other. I suggest you work as closely as possible with a good vet, and I sure do wish you the best of luck.