Depending on the actual nature of the illness, another possibility is congenital, which means a condition that is present from birth, whether due to environmental or genetic causes.
While not not all congenital conditions are incurable, some are, and if you are talking about an adult when you say they have a congenital condition, it strongly implies that it it is difficult or impossible to cure, because they've had it since infancy and still have it as an adult.
Congenital also can mean "inherent in one's nature" which implies that one is stuck with the condition and can't change it, as when someone is called a "congenital liar".