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Morendil comments on Beauty quips, "I'd shut up and multiply!" - Less Wrong
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My reasoning was a bit simpler.
Prior to the experiment, the probability of heads was 50%, tails 50%. Upon waking.. she learns no new information. She knew in advance she was going to wake up, and they tell her nothing.
So how could her beliefs possibly change?
One of the major take-aways I got from actually reading Jaynes was how he is always careful to write probabilities as conditioned on all prior knowledge: P(A|X) where X is our "background knowledge".
This is useful in the present case since we can distinguish X, Beauty's background knowledge about which way a given coin might land, and X', which represents X plus the description of the experimental setup, including the number of awakenings in each case.
That - the difference between X and X' - is the new information that Beauty learns and which might make P(heads|X') different from P(heads|X).