PA-44 Flight Instructor reported a runway excursion during landing after the aircraft veered to the left regardless of pilot input until the aircraft stopped in the grass. The Instructor stated the cause may have been a rudder connection issue.

Narrative

Everything started off as normal. We were crossing midfield to join the left downwind for Runway XX. After we joined the left downwind; we dropped our gear below 150 knots and proceeded to follow the traffic pattern as usual. As we were coming down on the short final; everything was looking good and we were holding centerline. It wasn't until we touched down where the aircraft started to veer off to the left. We both had right rudder in all the way. Power was all the way idle; and we were holding down the brakes as well; but the airplane kept going to the left. Once we knew; there was no way of getting it back to centerline we braced for impact. We went into the grass and approached the ditch that we ended up bouncing over and slightly missing. The plane was finally coming to a stop and we cut mixture off/got out. The FBO manager came over to try and assist us and we left the scene as it was. We did a little preflight afterwards just to see what had all been damaged. After assessing the damage we realized the vertical stabilizer was cracked; propeller had hit the grass; and the steering wheel bolt had been busted. After reflecting on the situation; I believe that there must've been a connection issue between the rudder and steering wheel. The rudder pedals were very loose after the runway excursion and the steering wheel bolt that connects them together was busted. So no matter how much rudder we used to keep it on centerline; the airplane was still steering to the left.