Combine water and kombu in a saucepan. Let stand until kombu is softened, about 20 minutes.
Bring water to a boil. Add turnips; simmer over medium heat until tender, about 10 minutes. Transfer turnips to 2 serving bowls. Place reserved leaves in the hot water; cook until wilted, 1 to 2 minutes. Divide leaves and cooking water between bowls. Discard kombu.
Combine miso paste, sugar, sake, and mirin in a small pot over low heat. Cook, stirring constantly, until thickened, 3 to 5 minutes. Pour over turnips in the bowls.