The pilot stated that during the engine run-up, prior to the departure from runway 31, the fuel mixture was reduced to clear the plugs. A second run up check was complete with no discrepancies noted. Just after the takeoff the engine started to misfire. The engine smoothed out momentarily and again began to misfire followed by the propeller stopping. The airplane made contact with power lines, which separated. The airplane then impacted onto a building's roof causing damage to the building before stopping. The occupants exited the airplane without assistance. The FAA representative stated that after the wreckage was recovered a post accident engine inspection and test run was completed. The inspection revealed that the fuel mixture control cable broke, which cut off the fuel at the carburetor. The engine ran without any discrepancies when tested. As per the owner of the airplane, the last inspection to the airplane was on October 3, 2006.