It sounds as through your daughter has not been able to stabilize yet from the surgery and balancing out her levels yet that work for her.
All the diet and exercise may not work until the body adjusts from the shock of removing the gland and not balanced out yet with the meds.
It can be a long process for some - even years until the body adjusts to the changes.
Right now your daughter and you need to keep an eye on keeping her TSH suppressed under 1.0 to limit recurrance of the Pap cancer. Once her body adjusts to a normal level of hormone from the meds she should start to feel much better.
Keeping a good diet is important - but exercise - if it is vigorous - may worsen her until she stabilizes.
Welcome to the site. How long ago did she receive her 2nd RAI treatment? Maybe that's what is making her tired right now? I'm sure some others who have been through that will know much more about it than I do.
As far as the weight gain, I can sympathize with that. I put on about 30 pounds over the course of a few months while waiting for my PCP to finally believe that there was something wrong with me and I was diagnosed with hypothyroidism in June 08, with diagnosis of Hashimoto's just last week. I have been on synthroid since June and am really struggling to get rid of the weight. I eat healthy and am very active.
For your daughter, I would first suggest that she needs to get all of her thyoid levels evened out, which may take a while. I would also suggest that you check out the Weight Loss and Dieting forum as there are a lot of good ideas there.
Good luck.