The flight instructor of the multi-engine airplane reported that, during a simulated single-engine landing, while the pilot-receiving-instruction was flying, during touchdown the airplane veered to the left. He added that he took control of the airplane "immediately" and added right rudder to correct, but the airplane exited the runway and the nose landing gear collapsed. The airplane sustained substantial damage to the nose landing gear structure. The flight instructor reported that there were no preaccident mechanical failures or malfunctions with the airplane that would have precluded normal operation. A review of recorded data from the automated weather observation station at the accident airport reported that, about the time of the accident, the wind was from 180° at 14 knots, gusting to 21 knots. The airplane landed on runway 19R.