I also have an 11 year old daughter and we too fell into the shouting/reacting pattern. It is impossible for her to change the situation, so it is up to you. I know that it is not easy, but start with little things.
Instead of shouting and fighting with her to get up in the mornings (a constant battle with my daughter), get into bed with her and tickle her awake and just snuggle with her. Chances are very good that she will react negatively initially, but keep on doing it.
Put little notes in her lunchbox. My nickname for my daughter in Nu-nu. I write her little notes like "Thinking of you Nu-nu" or "Hello Nu-nu, mommy loves you" etc.
What you can also do is to make special mommy-and-me time for her. Say, once a week/2 weeks take her out somewhere. Just the two of you. Take her shopping or go to a nice coffee shop for cake. If you are worried about what you two will chat about when there, then do what I do. We play a game called 'What was the: worst part of your week/ best part of your week/ funniest thing that happened to you during the week/ etc. Both of you have to answer these questions. If you know what kind of music she likes, buy her a CD without her having to ask you for it.
You know what her favourite food is. Announce to the whole family that say, Wednesday nights are your daughter's night to have her favourite food. Then make her favourite food.
If she and your 15 year fight a lot (which I guess they do), it usually is an unfair fight. One in which the 15 year old usually has the upper hand. Make sure to stand up for your little one. Make it clear to your 15 year old, in front of your 11 year old, that you will not tolerate ugliness directed at your little one.
Talk to your daughter, whether she wants to or not. When she is in her room, go sit by her on the bed and tell her about your day. She might not respond at all at first, but eventually she will feel that you have taken time out of your busy day to chat just to her.
Eleven is a very tricky age. A lot of developmental changes are taking place in their little bodies. She is confused, because she is too old to play with dolls (she still wants to, by the way), but too young to do the things her sister is doing. She is a child in a woman's body (almost). She feels sad/anxious/angry, but can't explain why. All these feelings are very overwhelming for her.
It is not too late to change your relationship for the better with her. Good luck.