basmelech: I read this topic to my sisters, and they said we should definitely give advice because we have the best sibling relationships ever, despite being about as different as genetically similar people could get.
First thing is, you need to realize that your sister is a person. That means she has her own perspective and opinions, and probably likes to talk about them. Ask her about them! Do you know who your sister's friends are? Do you know who her favorite teacher is and what she likes to do in her spare time? Ask her how her day went.
Second thing, is quantity time. We spend a lot of time together, even if I'm on the computer and they're on the phone, we're in the same room. That doesn't mean they know everything going on in my life. But they know when something is going on, and I know when they're upset about something, and we're there to talk if we want to.
Third thing - do something together. We go on walks and little trips to the park etc. a lot and can easily spend hours shmoozing about nothing. Sometimes we stay up late and blast music and dance together. (our parents do not approve of this)
Fourth - don't try to change them. We're all very different and we have to accept each other. I use the internet, she doesn't - that's great for her, and this is great for me. We do tell each other when something bothers us. My sister will tell me that she doesn't think a particular expression is nice to use, but she isn't judging me, just telling me her feelings. You want to be able to do that with your sister.
Fifth - realize that this isn't forever. Someday, you will live in different places and only be able to talk on the phone and visit. So value the time you have now, it is a gift!
Finally, you sound like a great sister, so don't worry. I'm sure she'll come to love spending time with you!