If done right, chopping off a head would be fairly efficient and humane - definitely better than electricity or hanging as other have mentioned.
That said, I remember reading about victims of the guillotine in France who would respond to questions for quite some time before death set in. Presumably, the brain continue to work despite the head being separated from the body for a time. Makes sense, but I'm not about to test it out.
I would think that being blown to bits by high explosives would be fairly quick and painless, but other than that, you're going to feel it.
As to the poor girl in question, it's not clear whether she killed the child or not. One thing is for sure. She got blamed for it and in Saudi, I would not ever want to stand before their legal system because the whole thing is a giant kangaroo court.