After such an incident, it's important to give her some space at first. But then, start with simple conversations. For example, when you see her, ask how she is doing. As time goes on, you can suggest doing something together like going to a movie or a park. Show respect for her feelings throughout the process. Try to understand her point of view and let her know that you value her as a family member. This will gradually help in building a better relationship.
Be honest and upfront. Admit if you made a mistake and show that you are committed to improving the situation. You could also find some common interests and start conversations around those. For instance, if you both like reading, talk about books.
One way is to do small favors for her. It could be as simple as helping her with a task she finds difficult or bringing her a small gift when you meet. This shows that you are trying to be friendly and make up for the uncomfortable incident. Also, involve her in family activities more often, so she can see your positive side and that you are an important part of the family.