Whisk milk and pudding mix together in a large bowl until smooth. Fold in whipped topping and set aside.
Pour coffee into a shallow bowl. Dip one donut into the coffee and hold for 5 to 10 seconds on each side. Place in the bottom of an 8-inch square baking dish. Repeat with 11 more donuts to fill the bottom of the dish.
Spread 1/2 of the pudding mixture over the donuts. Repeat the process with remaining 12 donuts and pudding mixture. Lightly sprinkle cocoa powder over top.
Cover and refrigerate, 8 hours to overnight. Serve chilled.