Pilot flying PA28 aircraft reported nose wheel steering failure during landing rollout; resulting in a runway excursion.

Narrative

I was conducting a solo training flight under part 91 with the goal of practicing takeoffs and landings in the traffic pattern. There was precipitation in the area and some light drizzle over the airport. I waited for the precipitation to move out of the area before I flew. There was no precipitation or adverse weather throughout the duration of the flight. Winds were from the south at 4 kts. Runup; taxi; and takeoff were normal. I made both left and right turns when taxiing to the Runway and did not encounter any issues with directional control during the takeoff roll (I would have rejected the takeoff if I had). I was flying a left traffic pattern; so I applied left rudder when turning from crosswind to downwind. When established in the downwind I released pressure from the left rudder pedal and noticed that the right rudder pedal did not move in the opposite direction. Upon further investigation; I noticed that when I pressed on the left pedal the airplane yawed to the left but the right pedal did not move in the opposite direction. I do not remember whether I attempted to press on the right pedal. I decided to make a full stop landing and taxi back to the ramp to have a mechanic inspect the rudder pedals. At this point I thought that the rudder was just behaving a little abnormally and did not anticipate any issues with landing and/or steering the airplane.The landing was normal. I did not land on the nosewheel or excessively side load the gear. After I touched down; I noticed that I was drifting to the left and applied right rudder to compensate. I ended up pressing on the right rudder pedal as hard as I could. The pedal did not move and the airplane did not respond. Realizing that I was unable to steer; I braked as hard as I could in an attempt to stop the airplane before I exited the runway. When I came to a stop; the nosewheel and left main were in the grass. I do not remember whether the right main was in the grass or on the shoulder of the runway. I shut down the airplane and requested assistance from the Tower.A Mechanic arrived and towed the airplane back to the ramp. He told me that my airplane did not leave skid marks on the runway. Nobody was injured. Neither the airplane nor any airport property was damaged by the runway excursion. The Mechanic opened the engine cowling and immediately identified the issue - a rod which connects the nose wheel to the rudder pedals had snapped; which would have disallowed the airplane from steering to the right. A CFI that I talked to believes that this could have been caused by a bad weld that was concealed by paint. In this time; the Mechanic did not notice any issues with the rudder. A final report from the Mechanic is still pending; but at this time with the information available to me; I believe that the failure was with the nosewheel steering only and not the rudder. Therefore; (and this is also the opinion of 3 CFIs that I talked to) since nosewheel steering is not a flight control system; this does not constitute a flight control system malfunction or failure which requires immediate NTSB notification. If further investigation by mechanics reveals issues with the rudder or other flight control systems; I will immediately notify the NTSB. In hindsight; I believe when I noticed the rudder pedal issue I should have taken more time to troubleshoot rather than attempt to land the airplane immediately. If I had taken more time here; I might have also anticipated issues with directional control on landing.