why is Annie is important to the conflict and overall plot of the play?

The Miracle Worker would end in despair if it were not for Annie Sullivan. While the story focuses on Helen Keller, a young girl who became deaf and blind because of a terrible illness when she was almost two years old, Annie Sullivan plays a principle role in the plot as Helen's teacher, also called a governess. Though many other adults, including Helen's mother and father, often lose hope that Helen will ever learn and/or behave as a normal child because of her disabilities, Annie has great faith in Helen. She suggests that Helen can learn to behave like other children; she faithfully believes that Helen can learn to communicate through sign language, and that sign language can be used to give her an education. At the time Annie met Helen (1887), there was little history of teaching blind and deaf people how to communicate. This play, based on the true history of Helen Keller and her teacher Annie Sullivan, shows how Annie Sullivan's confident hope in Helen's intellectual abilities led to miracles of communication and education in Helen's life.
Not only does The Miracle Worker focus on the conflicts that Helen faced and learned to overcome, but it also focuses on the struggles that Annie faced. Short portions of the story focus on Annie's past experiences, showing conflicts she faced while growing up, as well as conflicts that she continues to face as a young adult. For instance, readers learn that Annie and her brother, Jimmie, grew up as orphans in a poorly maintained almshouse. While there, Jimmie died of sickness; Annie felt responsible for keeping her brother safe and healthy. Annie was a young child when her brother died; notably, though Annie was not blind (like Helen), but her eyesight was severely impaired. In fact, Annie had to undergo multiple surgeries to attempt to correct her vision. Even while Helen's teacher, she had to wear dark glasses to protect her eyes from too much light. Her personal vision problems likely increased her sympathy and compassion for Helen, as a blind and deaf child. 
Because of the tragedy of her brother's loss, Annie states, "God must owe me a resurrection." In other words, because she lost someone so dear to her (her brother), she believes that God owes her another chance to love and care for someone. At the start of the play, Annie gets extremely frustrated with Helen's poor behavior. Helen throws tantrums where she demands her own way, such as when she insists upon eating with her hands. Annie refuses to give into Helen's tantrums, as Helen's parents often did, and insists on Helen learning to eat with silverware. It is not easy for Annie to enjoy being with Helen early in the play; it was difficult for her to love Helen. However, as time goes on and she continues to practice patience with Helen, she learns to love her. By the end of the play, Helen receives a miracle (learning to communicate through sign language), and Annie gets to love and care for someone once again. She is given her "resurrection." After Helen understands her first word in sign language, water, we read:

Annie: I, love, Helen.
[She clutches the child to her, tight this time, not spelling, whispering into her hair.]
Forever and -- . . . ever.

Annie is an important part of the conflicts of the play as she wrestles with Helen's strong will, teaching her to eat with silverware and to behave like other children. She also faces conflict as she opposes Helen's parents, expecting that Helen learn to behave with good manners and to communicate, when others do not think she is capable. Finally, Annie also faces conflict in her own life, learning to live with her impaired vision and to love Helen, after losing the brother she loved dearly. Without Annie's determination to face these conflicts, Helen may never have learned how to communicate.