What helped Helen in strengthening her kinship with the world?

During her childhood, Helen was increasingly alienated by her inability to communicate. What first increased her sense of kinship with the world was her ability to communicate using manual writing. This is what she means when she says she learned the "names and uses" of things:

The more I handled things and learned their names and uses, the more joyous and confident grew my sense of kinship with the rest of the world.

Additionally, Miss Sullivan took her out into nature amongst the butterflies and daisies, sat her on the "warm grass," and taught Helen the benefits of nature, which also increased her sense of kinship with the world.
Keller also tells a story about overcoming a fear. She had been left by Miss Sullivan for a few moments sitting in a tree when a storm came up and terrified her. This made her scared to climb trees, but a sense of kinship with a mimosa tree helped her overcome the fear:

Was there ever anything so exquisitely beautiful in the world before! Its delicate blossoms shrank from the slightest earthly touch; it seemed as if a tree of paradise had been transplanted to earth. . . . I kept on climbing higher and higher, until I reached a little seat which somebody had built there so long ago that it had grown part of the tree itself. I sat there for a long, long time, feeling like a fairy on a rosy cloud.

Later, during a trip to Boston, Helen Keller develops a sense of kinship with the world through reading. She primarily reads children's fiction at this point, and she says she did so without any critical awareness of whether a book was well-written or not. She simply loved and accepted the stories as she did the sunshine. She especially loved Little Women as a text that helped her feel kinship with children who could hear and see.

Given her serious disabilities, Helen often feels intense frustration at her isolation from the world around her. Yet Annie Sullivan is determined to help Helen engage with the world and greatly enrich her life-experiences. Annie knows, as Helen does, that the more new objects she can touch with her hands, the more joyous and confident she will become in her kinship with the rest of the world.
To this end, Annie often takes Helen on walks in the country. As they sit together on the banks of the Tennessee River, Annie gives Helen lessons on the natural world and all its joys and beauties. Helen learns how the sun and the rain make trees and plants grow; how animals such as birds, lions, squirrels, and deer live and thrive on the land. The more Helen comes to know about the natural world, the more she takes delight in it; and the more she takes delight in it, the greater the degree of kinship she feels towards her environment.