Yes Itachi died from illness, but kirin did have a part in it. Think about it:
If Itachi hadn't been fighting Sasuke that day and he had just been leisurely eating an ice cream cone or taking a walk during that time, do you really think that Itachi would have just randomly succumbed to the illness and keeled over and died? Probably not. The fight did push Itachi's body to the point of giving in to the illness.
If Itachi hadn't been fighting Sasuke that day and he had just been leisurely eating an ice cream cone or taking a walk during that time, do you really think that Itachi would have just randomly succumbed to the illness and keeled over and died? Probably not. The fight did push Itachi's body to the point of giving in to the illness.