You do not say anything about your partner's relationship with his child, or with his ex. A lot of us on ST eventually give up because partner/DH puts the child/ren before us, but it does not sound as though this is your biggest problem, just that you can't abide his daughter.

It's a very common feeling - SMs are expected to like or even love our SKIDS when there is absolutely nothing loveable about them, and they seem like a rival for our partner's love and attention. However, you do mention several times about not having your own children yet, and maybe you see this girl as a potential threat to any future family you may have with your partner.
She does sound a rather unsavoury, spoiled little girl with some not very nice ways, but she is part of his life, and unlikely to go away for the next dozen years or so. Only you can say if your love for your partner balances out the aggravation for you - this is a question of balance that a lot of us face, even those in stable relationships like myself, who find the step parenting role almost impossible at times.

Welcome, I am going to tell you from my experience that I think you should leave this situation. When I met my (now) DH his kids were a bit older (8 and 9)I liked them enough at first and they adored me and I got along OK with both (yes I have TWO) BMs. I would try to do lots of fun stuff whenever DH had them, would spend my money on them, took Karate lessons with them, would bring them to work with me (which they thought was fun) etc. however after the first "honeymoon" year, things slowly started to deteriorate. BMs started showing their true colors, I ended up getting in a giant brawl with BM#2 over nonsense that SS actually started and the whole skid "relationship" so to speak that was already hanging on by a thread started to unravel.

Then DH and I had a baby 2 years ago and the whole skid thing went to hell in a hand basket. I HATE both BMs and I really don't like skids at all. Over the years I have come to see them for what they are which is a carbon copy of their worthless mothers who only show an interest in DH when there is a gift giving occasion on the horizon or when they need money for something. When they do come over (which is about once a month or less now) I either leave or just do my own thing and stear clear of them. I, like you, could care less if I never saw them again. I could care less if my BS2 ever has a relationship with them, in fact I would prefer that he not. My skids are now almost 13 and almost 14 so at least I have an end in sight.

I guess the way I look at it is you are the same age that I was when I met my DH, if you want to have a child of your own, I would suggest that you NOT do it in the situation that you are in, as you will most definately regret it. I can tell you that your resentment and dislike for this child and her mother are NOT going to go away and in fact will most likely get worse and you have THIRTEEN years to go! Move on now, while you still have time to meet someone else who doesn't have kids.