In a medium saucepan, combine the rice, water, mirin, vinegar and sugar and bring to a boil. Cover and cook over low heat for 20 minutes. Fluff the rice. Transfer to a bowl and let cool. Season with salt.
Preheat the oven to 325°. In a large nonstick skillet, heat 1/4 inch of oil until shimmering. For each cake, pack the rice into a 1/4 cup measuring cup; unmold and flatten slightly. Cook the cakes over moderately high heat, pressing, until golden brown, about 4 minutes. Turn and cook for 4 minutes longer. Serve hot.