i am sorry. my daughter was like that as well. you can try different cute toothbrushes, try reading a book while someone else brushes, or let her brush yours while you do hers. im not sure if those things would matter at 11 months.
in reality for me, nothing worked so we had to hold her down. actually, when she would start crying, it was so much easier to brush. we hated doing it, but would hate more for her to have cavities. please do whatever you have to. my kids have no cavities, but everytime i go to the dentist, there are tons of 2 year olds getting fillings! one thing completely separate is dont give your daughter juice. if she never has it, she will never ask for it.
one thing you might want to try, every night go through the brushing routine, no matter how it plays out. on the nights when you didnt get a good brushing, go back and do it once she is asleep. that way you are still teaching her to do, and then are actually getting a chance to do it correctly.
although after i brush, they brush their own teeth, i personally wouldnt let a child brush their own teeth solely. my sister and SIL both did it thinking their 3 year olds were capable, and all the kids have tons of cavities. they just arent capable of doing a thorough brushing at a young age. the dentist actually told me that children under 6 really cant get everywhere.
all of a sudden, my daughter was fine. it was just like that, and she was fine ever since. i do agree to get the rotating ones for now, the brush does most of the work so its just easier. good luck.