In trying to assess a person's thyroid status. the most important step is an evaluation for symptoms that occur more frequently with hypothyroidism than otherwise. Does she have any symptoms other than weight gain? Also, how much weight loss, over what period?
Following that should be blood tests. Your daughter's high TSH and high TPO are indicative of Hashimoto's Thyroiditis. With Hashi's the autoimmune system erroneously identifies the thyroid gland as foreign to the body and produces antibodies to attack and, over an extended period, destroy the gland. As this is occurring the gland's output of thyroid hormone diminishes and the hypothalamus/pituitary system starts increasing output of TSH to try and stimulate more output from the gland. I cannot evaluate her other test results because I am not sure if they are Total T4 and Total T3, or Free T4 and Free T3. I think I know, but just to confirm, please post the reference ranges shown on the lab report for those two tests.
There is also a condition called postpartum thyroiditis which occurs more frequenlty when the patient has autoimmune antibodies. It manifests initially as hyperthyroidism, then progresses to hypothyroidism.
Hashimoto's Thyroiditis is the most common cause for diagnosed hypothyroidism. Both the TSH and TPO test are indicative of autoimmune dysfunction (Hashimoto's Thyroiditis). Along with those tests it would be expected that her thyroid hormone levels would be on the lower end of their ranges, resulting in hypothyroid symptoms: however, while her Total T4 was only at 4% of its range, her Free T3 was at 71% of its range. I don't understand why the doctor tested Total T4 instead of Free T4. Most all T4 is bound to protein and rendered biologically inactive. Only a small portion not bound to protein is called Free T4. Total T4 has become almost an outdated test. Maybe the lab person mistakenly identified the test to be done as Total T4. If her Total T4 is indicative of her Free T4, then that would be unusual considering her higher Free T3 level.
It is important to note that hypothyroidism is not just inadequate thyroid hormone, but instead, it is "insufficient T3 effect in tissue throughout the body, due to inadequate supply of, or response to, thyroid hormone. For that reason it is important to do additional testing. Since the doctor wanted to do followup tests, I highly recommend testing for Free T4, Free T3 again, Reverse T3, cortisol, Vitamin D, B12 and ferritin. The additional tests are all important for their effect on thyroid hormone. In addition, I think it is important that an ultrasound test of the thyroid gland be done to assess what condition it is in with the ongoing attack from the antibodies of Hashi's.
There may be other issues as well, but from the initial test results, I think she should get these done first, and then proceed from there.