During landing, the nosewheel began to shimmy while the brakes were being applied. The nose gear failed at the end of the runway, and the plane continued off the end and down an embankment, then nosed over. The pilot stated in part that he touched down past the numbers some 100 ft and had no left brake. When the nosewheel began to shimmy he started to go-around, but then the shaking became so bad that he aborted the go-around. A witness stated that he saw the plane on a steep final approach, but observed no flare-out. He reported that the plane touched down near mid-field in a 3 point attitude, but did not bounce. The passenger indicated that the plane landed with no flaps about 1/3 down the runway. He stated the touchdown was smooth and the nosewheel did not shimmy until the tires squealed from heavy braking. Examination of the brakes revealed no pre-crash malfunctions. The nosewheel shimmy damper was found broken from the nose strut with evidence of ground impact.