D.B.
Good for you for showing the maturity to realize that parents need help and little kids need role models. I imagine you will be very important to both children as they grow.
Here's what's really important to remember about 2 and 3 year olds: they don't play or interact all that much with others. They do a lot of what's called "parallel play" which means they sit on the floor next to another kid (whether 2 or 12 years old) and just do their own thing. That's where they are developmentally. So you can't expect a child that age to do a puzzle with your or play "Go Fish" (they don't know their numbers, can't hold the cards, and don't have the attention span). So if a child doesn't seem involved or seems detached, that's probably not related to anything you're doing wrong. You just have to be more patient!
So, yes, get down on the floor, and just play with some toys. Don't "teach" her how to do things, just do them. Put the different shaped blocks in the shape sorter, put the oversized puzzle pieces in their places, read a book out loud. Eventually, your niece will come over and join you. Don't make a big deal out of it - just turn the book so she can see it, or hand her a puzzle piece and say, "Do you want to put this one in?" Very low key, okay? If she turns around and heads over to her stuffed animals, it doesn't mean she doesn't like you.
The other thing you can do is try to understand that she's not going to be thrilled about this baby. This is competition for her, and it's a boring little crying/spitting up thing! Sometimes, 3 year olds regress and start acting very baby-like, and it's frustrating to see that. So, you can help by bringing your sister a clean diaper and saying, "Here, let me help!" Maybe you can hold the baby while your sister does things with the older child. Or, do the reverse: "Okay, sis, you feed Baby and I'll feed the big girl!" Then get your niece a piece of fruit or some cheese and crackers or whatever she is allowed to have. Maybe you can take your niece for a short walk in the stroller to get away from the crying and give your sister a break. A kid is often distracted by nature and people passing by, and forgets who's pushing the stroller. Even if she's crying to begin with, that doesn't mean she will stay that way. Sometimes you just have to ignore them and let them cry it out.
Your sister is learning how to juggle 2 kids, your niece is learning how to share Mom and Dad with a baby, and you're learning how to be a teen and a grown-up. Take it slowly.