Preheat oven to 350 degrees F (175 degrees C). Grease and flour a 9x13-inch baking dish.
Mix cake mix and pudding mix together in a bowl. Beat milk, vegetable oil, 5 tablespoons Key lime juice, and 1 tablespoon Key lime zest into cake mix mixture just until batter is smooth. Beat eggs, 1 at a time, into batter, beating until each egg is incorporated before adding the next. Pour batter into the prepared baking dish.
Bake in the preheated oven until a toothpick inserted in the center of the cake comes out clean, about 45 minutes. Cool cake to room temperature.
Mix lemon frosting, white chocolate, 1 tablespoon Key lime zest, and 1 tablespoon Key lime juice together in a bowl. Spread frosting over cake.