Hi Rita: Here are two Lady Baltimore Cake recipes that were previously posted here on the Recipe Link. I am wondering if either of these seem to be close to your grandmother's recipe.
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Lady Baltimore Cake
From: Fannie Farmer Cookbook

1/4 lb. butter
1 cup sugar
3 cups cake flour
2 tsp. baking powder
1/4 tsp. salt
3/4 cup milk
1 tsp. vanilla
4 egg whites

Preheat oven to 350 degrees F. Butter and lightly flour two 8-inch round cake pans.

Cream the butter until it is smooth, slowly add the sugar,
and beat until light and fluffy.

Combine the flour, baking powder, and salt; add to the first mixture, add the milk and vanilla, and beat until well blended.

Beat the egg whites separately until stiff but not dry. Stir 1/4 of the whites into the batter, then gently fold in the remaining whites.

Spread in the pans and bake for 25-30 minutes or until a toothpick comes out clean. Cool in the pans for 5 minutes before turning out onto racks.

Fill with Lady Baltimore Filling and frost with Seven-Minute Frosting.

Lady Baltimore Filling:

One recipe Seven Minute Frosting
1 cup chopped pecans
3 dried figs, cut in bits
1/2 cup raisins
1/2 tsp. almond extract

Divide the frosting in half. Fold the pecans, figs, raisins
and almond extract into half the frosting and use it as a
filling between the two cake layers. Use the remaining plain frosting to cover the sides and top of the cake.

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Lady Baltimore Cake
Source: The Household Searchlight

1 cup butter or butter substitute
1/4 cup sweetened condensed milk
3/4 cup water
4 tsp baking powder
1 tsp vanilla
3 1/2 cups cake flour
1/2 tsp salt
1 7/8 cups sugar
6 egg whites
1 tsp orange flavoring

Cream butter. Add gradually sugar, which has been sifted before measuring. Beat until smooth and creamy.

Sift flour, measure, and sift with baking powder and salt. Combine milk and water. Add alternately with dry ingredients to first mixture. Add flavorings. Fold in stiffly beaten egg whites. Pour into well-oiled layer cake pans.

Bake in moderate oven (375 F) about 35 minutes. Use fruit and nut filling between layers and a boiled icing for top and sides.