At about 10:59 EDT, the pilot of a Cessna 172RG lost control on landing at Kendall-Tamiami Executive Airport. Neither the pilot nor the passenger were injured. The pilot said that, as he climbed through 500 feet, the airplane made an uncommanded yaw to the left and the turn coordinator indicated a skid to the left. He returned for landing and, while trying to use the manual rudder trim, noted that the trim wheel spun freely. While on short final approach, the tower advised him to go around because the landing gear was not extended. He selected the gear down and landed, but the main landing gear was not fully down and locked at touchdown. Inspection revealed the chain for the rudder trim assembly was broken in the cockpit and the rudder trim was displaced fully to the left.