Asklemmy

By taking care of her. Take initiative, propose movies / games / ice-creams whatever. Things you like, things you think she'll like. She's having a hard time reaching out to you, do your best to reach out to her.

It's not your fault, but it isn't hers either. Try to have fun together, she'll get to know how you work and you don't one step at a time.

She’s a kid, you are hardly an adult. Just be kind, the things you are going through she can’t really understand at that age. And the things that a big deal to her, aren’t that big of a deal for you.

When you both are older and more mature you will hopefully just remember that you were kind and not the teenage drama that comes with adolescence and finding your place in the world.

It sounds like she’s constructed two competing versions of you in her mind—an idealized version that always understands and sympathizes with her, and a second version constructed from all the times you’ve failed to live up to those expectations.

If you can’t be her idealized version of yourself, you can demonstrate that you’re not the second version, either. Focus on proactively doing things for her when she’s not expecting you to—everything you do that doesn’t match what her mental model of you predicts you’ll do will weaken that model in her head.

Honestly, you might just have to wait until she's done with puberty. This just sounds like a typical teenager whose brain is addled with hormones. It'll die down with time.

Your sister sounds awful. What's your question, though?