Yes, she will need two passports.
She must enter and leave the US on the US passport and she'll need to show the UK passport when she arrives back in the UK to prove she has the right to live in the UK without a visa. Otherwise, how will they know she is a UK citizen and allowed to live here?
You wouldn't try to enter the UK without your ILR as proof of your right to live and work in the UK, and same goes for your daughter - she either needs UK passport or a visa in order to get back in.
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