In my opinion it would be far better to test for and use the biologically active
thyroid hormones, free T3 and free T4 to evaluate and dose your daughter, based more on her symptoms.
To better understand your daughter's situation, has the doctor determined the cause for her first TSH level, that was so high? Has the doctor tested for thyroid antibodies? If not, that would also be a good idea, because it can affect her treatment.