What you've written would work very well in a greeting card to your niece:
"I remember you as that little girl I used to dance with and be silly with and swim with. I've missed you so much and have wished for years, and wonder whether you'd like to reconnect?"
Then follow through with similar notes, perhaps for birthdays and holidays at first if she seems unsure about responding. Keep all your messages kind, positive, and non-critical. Don't push too hard, because she may have been given all sorts of reasons not to trust you. Give it time. Keep your heart and your arms open.
Be aware, too, that you may be asking for some real trouble, from your niece or her mother, by inviting her back into your life. If she needs counseling, you could become the recipient of some dysfunctional behavior (I've been in that situation more than once). Not saying you shouldn't go for it, though. You may be a real blessing to this young woman.