In a small heavy saucepan over medium-high heat, combine the sugar, 1/4 cup water, and the corn syrup and bring to a boil, stirring only a few times, until the syrup is clear. Reduce the heat to medium and simmer, without stirring, until the syrup is a deep amber, about 1 minute. Immediately pour the caramel into a 9-by-2-inch round cake pan or into 8-10 individual molds and tilt to distribute it evenly over the bottom and a little up the sides. Set aside.
Preheat the oven to 325°F In a large saucepan over medium-low heat, combine the condensed milk, half-and-half, whole milk, and cinnamon and bring to a simmer. Remove from the heat and let steep for 10 minutes. Remove the cinnamon and stir in the Kahlua and the diluted coffee. In a bowl, lightly whisk the eggs until blended. Gradually whisk in the warm milk mixture and then the vanilla. Strain the mixture through a fine-mesh sieve into the prepared mold(s). Place the mold(s) in a large roasting pan and put on the center rack of the oven. Pour boiling water to a depth of 1 inch into the baking pan. Bake, uncovered, until just set and a knife inserted near the center comes out clean, 1- 1 1/2 hours for the large flan, and 40-60 minutes for the individual molds. Remove the roasting pan from the oven, remove the flan(s), and transfer to a wire rack. LEt cool, cover with plastic wrap, and refrigerate for at least 6 hours or for up to 2 days.
To unmold, run a thin knife around the inside edge of the mold(s). Place a serving plate with a rim over the top and invert the plate and flan together. Lift off the mold carefully, scraping out any of the remaining caramel to run over the flan and around the plate. If you have made a large flan, cut into thin wedges to serve.