For the vanilla cake: Butter and flour two 8-inch round cake pans; line the bottoms with parchment rounds. Position an oven rack in the middle of the oven and preheat to 350 degrees F.
Sift together the flour, baking powder, baking soda and salt in a bowl. Set aside. Cream the butter and granulated sugar together in a mixer fitted with the paddle attachment on high speed until light and fluffy, about 2 minutes. Beat in the eggs one at a time, then the egg whites. Beat until smooth, about 30 seconds. Beat in the vanilla.
Reduce the speed to medium and add the flour mixture in 3 parts, alternating with the buttermilk and beginning and ending with the flour, making sure to scrape the sides of the bowl several times. Beat on high to smooth the batter, 10 to 15 seconds. Remove the bowl from the mixer and fold the sprinkles in by hand.
Divide the batter between the 2 pans and smooth the tops with a spatula. Bake until a tester inserted in the centers comes out clean, 30 to 35 minutes. Cool the cakes on a rack for about 15 minutes, then remove from the pans, peel off the parchment and cool completely.
For the key lime buttercream: Cream the butter and cream cheese in a clean mixer bowl in the stand mixer fitted with the paddle attachment until light and fluffy, about 2 minutes. Add the confectioners' sugar and salt and beat on the lowest speed until combined. Add the vanilla, lime zest and 2 tablespoons of the juice, then slowly increase the speed to high and beat until very light and fluffy, 1 to 2 minutes, adding the last tablespoon of lime juice if the buttercream seems too stiff to spread easily.
Set one cake layer on a serving platter or cake turntable if you have one. Spread the top with about 1 1/2 cups buttercream almost to the edges. Top with the other cake layer, flat-side up. Spread the top of the cake with a heaping cup of buttercream and smooth with a large offset spatula. Spread the remaining buttercream on the sides of the cake. Hold the offset spatula at a 45-degree angle to the cake and rotate the cake (or turntable) to smooth the edges, leaving a thin layer of buttercream with the layers slightly showing through.
Sprinkle the top of the cake with sprinkles and decorate the edge with the lime slices. Chill the cake to set the layers for at least 30 minutes or up to 1 hour before serving.