I LOVE Being an aunt, but with each relationship I have with a kid, I also have a well defined relationship with the mother. I am the natural aunt to one niece, age 8, an Auntie-Cousin to two more, 10 and 5, and Auntie by choice to five more, 14, 8, 5, 3, and 1. For the most part, I'm only talking the relationship with my natural niece, my sister's daughter. First, As much as you want to be all around, the mother will be hesitant to leave the baby with anyone but herself. I promise it has NOTHING to do with you as a person. By the time a second or third kid comes, they're happy to hand the kid over for a good night's sleep, but the first one they will be wary to tear away from.
Secondly, as they are toddlers, you seriously only need like three toys, one of which has balls with it. I have a toychest full of toys she never played with, and she played with only one item for like three years. On the upside, I had a pleasant conversation with my niece when I asked her if I could donate all of the Doc McStuffins stuffed animals I had, and she said she was happy to have it donated if it goes to someone who deserves it. I promised her it would go to a little girl who loved Doc.
I also find that as they get older they get more awesome to hang out with. Last year I started an annual christmas shopping day with my niece, just her and me, and while I naturally forked out my credit card for her purchases, they were thoughtful and modest without my prodding. I also love taking her to movies, now that she doesn't sleep through them. My sister also set up a way for my niece to call me and her grandmother on her tablet whenever she wanted, but with my sister's ability to note when its happening. My niece knows that she can share things with me, but if I feel I need to, I will have a discussion with her mom on things she needs to know...