I'd imagine that all their lives, they've been told stuff like 'If you have faith the size of a mustard seed, you can move mountains' and also 'God is all-powerful and all-loving'.
They have faith. They trust that God will heal. From everything they've been told about the religion, praying a disease away is a piece of cake. Jesus did it with leprosy, so why shouldn't they be able to do it to? Giving her insulin would have shown doubt, and they can't have that when they're desperately trying to prove their faith. Plus, it's not like an all-powerful and all-loving god would let an innocent girl DIE because of that, right? But even if he did, then that's all part of his master plan.
They already have a thought-terminating cliche for every red flag their brain raises. It's truly awful what happened and that people would sooner lose a child over losing their religion.
Edit: Just got back from the article and a few followups. Nevermind! This was a fucking cult. >The Saints—a tight-knit group that only has 23 members in total, spread over three families.
The mom was sent to jail after having put her in coma just a couple of years before. She was then. Released and allowed to return home to finish the job. Horrible.