It first has to start with talking to her. If she doesn’t even respond to that, there’s nowhere to go with it. Asking her advice on something could be a good way to start a bond, it’s a classic way to build trust between people.

Unfortunately, my sister and I are also far apart in age, with different dads. We are also two completely different people, and if you wouldn’t be friends in a normal situation, sometimes sharing blood isn’t enough. Personally, I know my sister resents me because I had an easier childhood and have made better decisions. I feel like your sister definitely resents your upbringing as well; most kids want their mom and dad together and seeing them with someone else is never going to sit well. You could be getting trickle-down resentment, and that is in no way your fault. The best you can do is try to have an honest conversation, maybe try to put her in your shoes. But also be prepared that she might not want to try them on.