Combine first 11 ingredients in a stock pot. Bring to a boil, lower heat and simmer for 1 hour covered. Combine ground beef, chicken, sausage in a saucepan. Cook until done and add to tomato mixture. Simmer for another 20 minutes.
Layer bottom of a 13 by 9 by 2-inch pan with sauce. Add a layer of angel hair, pinch cream cheese over pasta, place a single layer of Swiss cheese, then 1/2 of cheddar and Jack cheese, repeat layers ending with sauce. Bake in a 350 degree oven for 30 minutes. Top with remaining cheese. Return to oven and continue to cook until cheese is melted. Cut into squares before serving.