Preheat the oven to 350 degrees F. Grease a 13 by 9 by 2-inch rectangle pan with 1 teaspoon of the butter. In a medium-size nonreactive saucepan, melt 2 tablespoons of the butter. Add the brown sugar, persimmons, and the lemon juice. Saute the mixture for 2 to 3 minutes, or until softened. In a small bowl, dissolve the cornstarch in the water. Add to the persimmon mixture and cook, stirring, until it thickens, about 2 minutes. Remove from the heat and let cool completely. In a large mixing bowl, cream 1/4 pound (1 stick) of the butter and the light brown sugar with an electric mixer. Add the eggs 1 at a time, beating well after each addition. Into a medium-size mixing bowl, sift 3 1/2 cups of the flour, the baking powder, baking soda, 1 teaspoon of the cinnamon, and the salt. Add the flour mixture, buttermilk, and vanilla to the butter mixture and beat with the electric mixer until everything is incorporated. In another medium-size mixing bowl, combine the remaining 1/4 cup (1/2 stick) butter, the remaining 1/2 cup flour, the remaining 1/2 cup brown sugar, and the remaining 1/4 teaspoon cinnamon. Using your hands, combine the mixture until it resembles fine crumbs. Spread half of the batter in the prepared pan. Spread the persimmon mixture over the batter. Drop heaping spoonfuls of the remaining batter over the persimmon mixture, about 1-inch apart. Sprinkle the crumb mixture evenly over the surface. Put the cake pan on a baking sheet and bake until golden brown, about 30 to 35 minutes. Remove from the oven and let cool slightly. In a medium-size mixing bowl, whisk together the confectioners' sugar, dark Karo, and milk until smooth. Drizzle the frosting over the top of the coffee cake. Cut into slices and serve warm.