Try this:
I also have a very headstrong daughter and this is what worked with her....Giving her choices. Many times children feel like they have no contorl over their lives, mom or dad tell them what to wear, when to eat, what to eat, etc. By giving the child choices, they feel like they are in control (when actually, you still are!). The key is to give 2 choices with each question (and ask as many as possible to give them more choices), either choice you are happy with.
Example:
When your daughter tells you she is thirsty, you say:
Would you like the red cup or the orange one?
Would you like juice or milk?
Would you like a lid or a straw?
Would you like to drink it in the kitchen or at the dining room table?
This works with discipline too...
When your daughter shows inappropriate behaviour, you say:
That behaviour is not acceptable in our home (or at church, or at the supermarket, etc.), would you like to spend 5 minutes time out in on your bed or in the dining room chair? If the child refuses to choose, you just up the time and repeat the same question. Again, if they refuse to choose, you just up the time and repeat the same question. Eventually they will see that the time is getting longer and longer and will choose.
The tough part is sticking with it in the beginning. When they refuse to choose (when it's not discipline) or try to give another option, you choose for them and STICK TO IT. This teaches them that if they won't choose from your options that you will take control back and choose for them. It won't take very long until they will choose because they will learn that if they don't you will choose for them and take the control away. If they refuse to choose (when it is discipline) you just keep upping the time and at some point you just stick with it so they will remember the next time that the punishment is worse if they don't choose.
My daughter is now 18 and a wonderful young lady, however, at 2 she was driving me crazy because she was so stubborn and strong willed. This method worked wonders on her. These children won't do what you want them to do just because you want them to do it, they won't do it until they decide to do it and that can be quite frustrating!
Good luck!!!