The certificated private pilot of the Piper PA-32-301T was performing takeoffs and landings on runway 29 to maintain her currency. During the third landing, the airplane departed the right side of the 75 foot-wide runway. The pilot reported that the rudder pedals were mushy after touchdown, during landing roll out, and that the airplane did not respond to her inputs to the left rudder pedal. However, the left rudder controls and nose wheel steering were operational when the pilot commanded the airplane to turn hard left to return to the runway pavement. Impact with the runway edge collapsed the nose gear. The pilot reported a total flight time of 464 hours, including 364 hours in the accident airplane. The reported winds at the time of the accident were from 040 degrees at 10 knots. Examination of the airplane by a Federal Aviation Administration (FAA) inspector, and a certificated mechanic with Inspection Authorization (IA) revealed that the airplane received substantial damage to the nose gear, firewall, propeller, and fuselage. Neither the FAA inspector nor the IA identified any pre-existing mechanical deficiencies with the airplane's steering, brakes or flight controls. On a previous flight in the same airplane three months prior to the accident, the pilot reported that the "nose wheel steering seemed squirrely as the airplane veered to the right and rudder pedal inputs were mushy." After that flight, maintenance on the airplane included replacing the left brake pads and tightening the nuts securing the nose wheel. Other pilots who recently flew the accident airplane reported nothing unusual with the nose wheel steering, and that the airplane was a "handful" during landings.