I'll second Portia and aeh above - your dd might not understand other people's feelings.
"she doesn't show any care about the feelings of people in her life. She orders people around and is basically a bully."
I had this issue with my child, or at least similar. it was hard to fathom, for me, anyway, if I was MORE firm with her, she would soften, but that is what occurred. I was at a point where I would respond to her screaming and rudeness by backing away and giving in, more and more, thinking she would "come around" or at least stop hollering. Or behave like my son, who just follows rules.
It literally took only 1 solid consequence to get her to get on a much better path, and it's been mostly uphill since then. I had to sit in her room for about 1.5 hours listening to her scream and not 'give in'.
While this might work for a neurotypical child, it's not likely to work if the underlying issue is a lack of understanding other people's reactions and emotions.
I'll also add for the OP - whether or not your dd can use help with social skills etc, there's a bit of a universal thing here that might help to look at. These are just my thoughts, and definitely aren't meant to minimize anything you're going through with your dd. Just food for thought. I think it's almost universal that most of us, going into parenthood, have a preconceived idea in our head of what it will be like or what we want our relationships with our children to be like. I think it's also probably almost 100% guaranteed that none of our children or our family dynamics once we have children, will turn out the way we dream about them unfolding. What makes a meaningful relationship with anyone (child or otherwise) isn't finding someone who is what we want them to be, but finding a way to appreciate who they are. That can be tough with children sometimes, simply because there's so much else entangled in our relationships with them - hopes, dreams, worries over the future, wanting to do the right thing, getting through the details of everyday life. Plus kids will be kids - issues with social skills etc aside

I have three children, and one is a lot like your dd. Another one is a lot like *you* (way more than I ever will be lol!). The third is just entirely a different person from the other two, who are entirely different from each other. None of them turned into the type of person/relationship you were hoping to find with your dd, but otoh, they are all three of them amazing people. Our lives as a family, the life they have grown up with so far, is very different from the life I'd imagined once upon a time - but it's *beyond* wonderful. It's not wonderful because my kids are angels - remember, I have one who is a lot like you describe your dd, and the other two, although different, come with their share of challenges. It's wonderful because they are unique, and in living through all of our times - happy times, fun times, and rough times - we've grown to appreciate what's wonderful in each of us. OK, we parents probably appreciate it more than our kids at this point in time... but that's ok, they're still kids

So my advice is - if you think that perhaps your dd needs help with social skills, look for professional guidance. Whether or not you think it's worth pursuing, google Aspergers + adolescent girls too, and see if any of the descriptions tend to fit your dd - if you see something there, seek professional guidance. And no matter what happens with all of that, try to let go of what you were looking for - that close relationship where you meet on common mind-ground - and instead make a conscious effort to spend time with your dd doing what she enjoys. It doesn't have to be a lot of time, just make sure it happens a few days each week, even if only for 15 minutes. It might not be all that much fun for either of you at first, but make it a priority and do it. I'd also spend a little bit of time (just a few minutes a day) either thinking about or journaling about the things that you find delightful, fun, interesting, or simply amazing about your dd. Smile with her. Make an effort to consciously love her for who she is (I am not implying at all that you don't love her as she is already - just saying use love as a verb, and focus on it).
If you can manage to let go of your expectations, you will most likely eventually find a relationship even more amazing and fulfilling than what you'd hoped for.
Best wishes,
polarbear
ps - also wanted to add - your dd is 10. She's about to head straight-on into what can be the most challenging time for bonding between parent and child. Adolescence, puberty, peers etc - the ages of 10 - 14 are among the toughest for most of the parents I know - so if nothing else, so have patience with your relationship and with your dd. Also reach out among your dd's friends' parents or other parents you know of same-age kids in real life, just as you've done here. You'll at least find good company
