The pilot reported that the aircraft was flared normally for the night landing, but it dropped suddenly and struck the runway heavily. Damage was caused to the nose gear strut. The pilot, believing he was losing control of the aircraft, carried out a go-around. During the subsequent landing, the pilot was unable to steer the aircraft which veered to the left and struck a cone marker before being brought to a stop. The aircraft had been observed to fly a close base leg followed by a steep final approach path. The pilot had misjudged the landing flare and during the subsequent heavy landing the nose gear strut was pushed upward through the aircraft nose, dislodging the windscreen and disconnecting the nose wheel steering. Unknown to the pilot, the propellers also contacted the runway and the tips of all blades had been bent.