But it does seem a bit over-the-top for your niece to be so very angry about your forgetting her birthday. I could see her being annoyed, and maybe having her feelings hurt a little, but a friendly sincere apology from you along with the belated birthday present ought to be able to clear the air.

My advice is to gently express your apology, to tell her how much her friendship means to you, to give her the present, and go from there. If she accepts this, you could take her out for a casual meal or treat of some sort. If she doesn't accept it, she's going to have to take some time to figure out some other way to deal with disappointments.

Apologize for forgetting and let her understand that people forget sometimes, which is no reflection of what she means to you. I would let the whole thing drop after that. She is ten years old and this will not change the way she feels about you once her ten year old brain forgets too. If you let it lie it will die, keep bringing it up and she will remember. IMO