Battle of the wills

Each of my kids can have a strong will, but Addison probably has the strongest will of each of them. Part of it is probably the age, but part of it is her. Today, she hit her brother because he was playing the piano and she wanted to play it all by herself. I told her she needed to say sorry and she refused. I told her if she didn't say sorry, she would be put in a time out. I counted to ten to give her a chance to say sorry first and she shook her head no the entire time. I put in her crib (I'm so not ready for a big girl bed for her) for her time out. She can now climb out the the crib on her own (which is why I need to put her in a big girl bed), and we had a battle of me putting her in a time out. If I stayed close, she stayed in her crib. if I was in the other room, she'd climb out. I kept asking her if she would say sorry and she would tell me no. We've had problems with her hitting her brother lately so we really need to get this to stop. After about 20 minutes, she finally said she'd say sorry. We we able to get out and give her brother's a hug and say sorry. She's been great since then. Sometimes that little girl wears me out. We've had many a battle of the wills lately, but I have hope!
We're working on being polite and using our manners and she's done great with saying please and thank you for the most part. I can't believe my little girl is growing up so fast!