The pilot reported she checked the weather prior to departing on the cross country flight and it appeared that an area of icing had cleared out of the area near the destination airport. While descending through 4,000 feet near the destination, the airplane began to pick up ice quickly. The pilot advised air traffic control and cycled the deice boots several times. The pilot reported she was unable to determine how much ice was accumulating on the airplane and she made the landing approach using "normal" approach speeds. The airplane touched down hard during the landing resulting in substantial damage to the internal structure of the left wing. The pilot reported that after the landing ice was visible along the leading edge of the wings. She stated she should have maintained a higher airspeed during the approach and landing.