Just a guess, but I'm thinking this is the recipe you are looking for. And you are correct, there isn't any fruit or nuts in the cake or frosting. I've not seen a recipe for Divinity Cake, but when I read your post I instantly thought this might be what you are looking for. The recipe I have does call for an Orange Filling, but it is easily omitted.

Ambrosia Cake:

1 cup butter or margarine, softened
2 cups sugar
4 large eggs
3 cups sifted cake flour
2 1/2 teaspoons baking powder
1/2 teaspoon salt
1 cup milk
1 teaspoon vanilla extract
1 teaspoon butter flavoring

Beat butter at medium speed of an electric mixer until creamy; gradually add sugar, beating well. Add eggs, one at a time, beating after each addition. Combine flour, baking powder and salt; add to butter mixture alternately with milk, beginning and ending with flour mixture. Mix after each addition. Stir in flavorings.

Pour batter into 3 greased and floured 9 inch round cake pans. Bake at 350' for 20-25 minutes, or until a wooden pick inserted in center comes out clean. Cool in pans on wire racks 10 minutes; remove from pans, and let cool completely on wire racks.

Divinity Frosting:

1 1/2 cups sugar
1/2 cup water
1/2 teaspoon cream of tartar
3 egg whites
1/2 teaspoon vanilla extract

Combine first three ingredients in a heavy saucepan. Cook over medium heat, stirring constantly, until mixture is clear. Cook without stirring, until mixture reaches soft ball stage, or candy thermometer registers 240'.

Beat egg whites at high speed of an electric mixer until soft peaks form; continue to beat, slowly adding hot syrup mixture. Add vanilla. Beat until stiff peaks form and frosting is spreading consistancy.

This particular recipe is counting on you using the orange filling and claims the frosting is enough to frost a three layer cake (top and sides). I'm thinking if you are omitting the Orange Filling and using the frosting alone you might want to double the recipe or make two batches. Just a thought. I hope this is what you are looking for and I have a feeling it is. Good luck and Happy Holidays!