Have all ingredients at room temperature. Preheat oven to 350 degrees. Grease and flour 3 8-inch by 2-inch round pans or line the bottoms with wax or parchment paper.
Sift together twice the flour, baking powder, and salt.
Combine milk and vanilla.
In a large bowl, beat butter until creamy, about 30 seconds. Gradually add 1½ C sugar and beat on high speed until lightened in color and texture, 3 to 5 minutes. Beat in egg yolks one at a time. Add the flour mixture in three parts, alternating with the milk mixture in 2 parts, beating on low speed or stirring with a rubber spatula until smooth and scraping the sides of the bowl as necessary.
In another bowl, beat egg whites and cream of tartar on medium speed until soft peaks form. Gradually add ¼ C sugar, beating on high speed, and continue beating until the peaks are still but not dry.
Use a rubber spatula to fold one-quarter of the egg whites into the egg yolk mixture, then fold in the remaining whites. Divide the batter among the pans and spread evenly. Bake until a toothpick inserted into the center comes out clean, 25 to 35 minutes. Let cool in the pans on a rack for 10 minutes. Slide a thin knife around the cake to detach it from the pans. Invert the cake and peel off the paper liners, if using. Let cool right side up on the rack.
Meanwhile, prepare frosting:
In a mixer, cream the first three ingredients. Gradually add the powdered sugar and mix well. Add the Kahlua. Dissolve the espresso in the water and add. Mix well.
Dissolve some (maybe a tablespoon) espresso in about 1½ cups water and put in a squeeze bottle.
Put one layer down (bottom side up, I think so the cake will absorb the liquid better) and squeeze about one third of the water-espresso mixture on the cake. (I started in the middle and went around in a circular pattern.) Frost that layer. Repeat with the next two layers and frost top and sides. Then with your hand, pat the toffee sprinkles around the sides. You can use some of the leftover frosting to pipe a design around the top edge. I think we had a bit of frosting left over even after that.
You might try a yellow cake box mix instead of the above recipe.