Amelia's Stubborness and some other randomness

Amelia is becoming increasinly stubborn, something I never really was, so it amazes me that someone so stubborn could have come out of me!
She has this thing with biting people, it's really quite painful now she has all those teeth. Anyhow, so sometimes she bites Joshua and I tell her to say sorry and give him a kiss, and she just outright says "NO!" The first time I heard that, I just kind of look at her stunned! My daughter said an outright no?! She knows why she shes saying no too. This is not good!
I kept asking her to say sorry, and every time she would looked at me and defiently say "NO!" Then I began getting very stern with her. "Amelia, say sorry to Joshua!". Still with defiance she said...."NO!"
Fuff, this was getting too much for me. Amelia, you're going to sit here until you say sorry to Joshua! So I sat her down on the bed, and pretty much left her there as she screamed and cried away!
Me: "Are you going to say sorry to Joshua now? If you say sorry you can get off the bed."
Amelia: Glares at me and continues to throw her fit
Me at a loss as to what to do: Look Amelia, see those marks on Joshua's arm? You gave him an ouchie. It's not nice. Say sorry please.
Amelia: "No sorry Joshua!"
Me: Ok, we're leaving you here.
So off I go with Joshua to get a snack for him, and I hear her crying away in my room, still throwing her fit. 2 minutes later I go in and ask her if she'll say sorry. She gets up, and very begrudingly says sorry to Joshua.
I turn around, pleased that I had stood it out, and glad Amelia had finally given in. I look back at her and shes looking at Joshua with a look of supierority and disdain, and she says "Sorry" again, but this time in a very "I'm am so much higher and mightier then thou, you should be thankful I graced you with my apolgy voice." Sheesh! Talk about pride!