@Letty,
Letty is absolutely right. I would use "wailed", too, unless I intended to convey that the child was over-reacting and being too infantile.
In common use, if we say that a human is "whining", [dogs whine, but they can't help it] we are making a judgement - we are implying that the utterance is persistent, annoying, excessive and silly, that the complaint or protest is trivial or childish, that the whiner is making too much of a fuss, that they ought to have kept their mouth shut. "Wailing", on the other hand, is merely descriptive of high-pitched sustained unhappy crying out.
Brave people accept their fate, cowards whine about it.