The pilot reported that after touchdown on runway 13, the airplane encountered a wind gust, and the airplane subsequently climbed about 5 ft above ground level (agl). The pilot applied back pressure on the yoke to regain a slight nose-high landing attitude when the airplane landed hard and bounced again to about 3 ft agl. Due to the airplane’s airspeed being below the single-engine best rate of climb airspeed, the pilot did not apply power for a go-around and continued the landing. The airplane landed hard again, the airplane’s nose landing gear collapsed, and the right main landing gear tire blew out. Unable to maintain directional control, the airplane exited the right side of the runway and came to rest upright in the grass. The airplane’s fuselage sustained substantial damage. Prior to landing, the pilot used on-board weather information to determine that the wind was from 200° at 17 knots, gusting to 26 knots. The airport’s runway 22 was listed in poor condition with a remark that the asphalt was cracked and had depressions. The pilot chose runway 13 for the improved surface condition and wider runway, which placed the runway crosswind component between 16 to 24 knots. The pilot reported that there were no preaccident mechanical malfunctions or failures with the airplane that would have precluded normal operation.