I was your niece except the new child was my own niece - big age gaps in my family.
I was nine when she came along and was the baby. I didn't mind her at all at first. In fact, until we got to the ages of your niece and your daughter, I hardly noticed her at all. I wasn't interested.
But as a toddler she started to annoy me - demanding attention, having tantrums, breaking my toys not playing nicely, picking her nose and developing an obsession with poos and bums.
These are normal things for a small child but I was a child myself and was also behaving normally.
We just didn't really spend a lot of time together until the phase passed because we wound each other up. It didn't take that long. By the time I was 15 I was back to ignoring her. Now we get on more like friends than aunt and niece.
You're lucky in the way that my parents and my brother and SIL were because we weren't under the same roof. They could keep us from antagonising each other until it passed. It would be far more difficult if they were siblings.
I don't think your niece, who is a child too, needs to be punished more than saying: 'Hang on, that's not nice' unless she actually harms your daughter.
You also have to take on board that three year olds are quite capable of deliberately winding people up - though they are also children and so it is also not their fault.