Assemble drip coffee maker according to manufacturer's directions. Place the ground coffee in the coffee filter or filter basket. Add cold water to coffee maker and brew to make 1 1/2 cups. Combine the brewed coffee and 2 tablespoons Kahlúa in a shallow dish, and cool.
Combine cheeses in a large bowl. Beat at high speed of a mixer until smooth. Add sugars and 2 tablespoons Kahlua, and beat until well-blended.
Split ladyfingers in half lengthwise.
Quickly dip 24 ladyfinger halves, flat sides down, into coffee mixture; place, dipped sides down, in the bottom of an 8-inch square baking dish, slightly overlapping ladyfinger halves. Spread half of cheese mixture over ladyfingers; sprinkle with 1 teaspoon cocoa. Repeat procedure with remaining ladyfinger halves, coffee mixture, cheese mixture, and 1 teaspoon cocoa.
Place 1 toothpick in each corner and 1 in the center of tiramisu (to prevent plastic wrap from sticking to cheese mixture); cover with plastic wrap. Chill 2 hours.