For the cake: Preheat the oven to 350 degrees F. Butter three 8-inch round cake pans. Line the pans with parchment paper, then butter the parchment. Dust with flour, tapping out the excess.
Whisk together the all-purpose and cake flours, baking powder and salt in a medium bowl. In the bowl of a stand mixer, beat the butter and sugar on medium speed until pale and fluffy, 3 to 5 minutes. Add the egg whites 1 at a time, beating well after each addition and scraping down the sides of the bowl as necessary. Beat in the vanilla. Reduce the mixer speed to low and gradually add the flour mixture in 2 batches, alternating with the milk and beginning and ending with the flour. Beat until just incorporated. Fold in the sprinkles.
Divide the batter evenly among the pans and bake the cakes until golden at the edges and a toothpick inserted into the center comes out clean, 25 to 30 minutes. Transfer the pans to wire racks and cool 15 minutes. Turn the cakes out of the pans and return to the racks to cool completely.
For the frosting: While the cake layers are cooling, whisk together the sugar and egg whites in a heatproof mixing bowl set over (not inside) a pan of simmering water. Whisk until the mixture is warm and the sugar is dissolved (it should feel completely smooth when rubbed between your fingers), 2 to 3 minutes. Transfer to the bowl of a stand mixer fitted with a whisk attachment.
Whisk the egg white mixture on medium speed for 5 minutes. Increase the speed to medium high and whisk the meringue until stiff, glossy peaks form and the bottom of the mixing bowl is cool to the touch, about 7 minutes.
Reduce the speed to medium low. Add the butter a few tablespoons at a time, beating well after each addition. (If the frosting appears to separate after all of the butter has been added, simply increase the speed to medium high and beat until smooth again.) Add the vanilla and salt and continue beating until the frosting is smooth and fluffy, 3 to 5 minutes.
Trim the tops of the cooled cake layers with a knife so they are flat. Place the bottom layer on a cake stand or platter and spread evenly with about 1 cup of the frosting. Place the second layer on top of the first. Spread evenly with about 1 cup frosting. Place the remaining layer on top of the second layer. Spread the remaining frosting over the top and sides of the cake, using an offset spatula to create large swirls. Decorate with sprinkles as desired.