Grease a 9x13-inch pan. Place puffed wheat in a large heat-proof bowl; set aside.
Bring corn syrup, white sugar, butter, cocoa powder, and brown sugar to a boil in a heavy saucepan. Remove cocoa mixture from heat; stir in vanilla extract. Pour cocoa mixture over puffed wheat cereal; stir until well combined.
Press puffed wheat mixture into prepared pan. Allow to cool completely before cutting into bars.