Could my 11 month old get an adjustment to her status if her visa expired?

I am a US citizen. My daughter is a British citizen. She entered the US on an ESTA in June. I immediately began applying for her US citizenship papers and have been given the run around by the local UCIS office, had MY citizenship come under question and been told to file at least 3 different types of applications. I have taken her to the local airport Immigration office to get an extension and been told no and that she needs to return to the UK prior to the ESTA expiration. 8 months later and still no US citizenship papers. My questions are: 1.) Is she not a naturalized citizen by birth? 2.) Are we outside the law at this point? 3.) If we are how much trouble are we in and what can we do to fix that? 4.) Are we able to travel to the UK to see her father? He is British in the military.