My experience with GoT/ASOIAF is somewhat limited; I have read the first two and a half books and I have seen I think a total of two episodes of the show. (Weirdly, I have all the seasons on DVD and the books on my shelf; one day I'll catch up.) That said, I do remember Daenerys very vividly.
I think you have some parts of her down pat. She has a certain arrogance and a certain "blood of the dragon" thing going on, and that's awesome. I love that she keeps making reference to her dragons, as they're clearly extremely important to her. However, there are some other parts that seem to be missing.
Though her arrogance is there, I feel like it's a shadow of the Targaryen pride I remember in the books. She seems far more passive than the Mother of Dragons should be. She's stuck somewhere without her dragons and she's not getting answers, and it seems like she's missing the pride and authority that belongs to her in canon.
What also stands out to me is her mental state. She seems to be fairly rational and acting like someone who is in complete control of their mental faculties. And while it's clear that Daenerys in her canon is extraordinarily capable, she is also a Targaryen and the madness of the Targaryen family is pretty legendary in GoT/ASOIAF canon. As I remember, even where I left off (which was admittedly a couple books ahead of where you take her from), she was starting to go kind of cuckoo. And what I mean by that is more of a megalomania than anything else; it's crazy inspired pretty heavily by pride.
Overall, I'm excited to see more out of Daenerys. She's a challenging character to play and I'm really excited to see her in game. These are just some things I've noticed and some things to keep in mind going forward.