The only time one of my children had been ill was like that.
She was comparing the way one of her children had been ill to another child when she says like that as a reference to her own past experience.
I'm not sure whether she means that only one of her children had been ill before, or that more than one of them had been ill before as it is worded very loosely and could be interpreted in a few different ways. Is she stating that she has more than one child or that more than one of her children has been ill before when she says 'one of my children'?
What do you think her meaning was?