C-172 Pilot reported an asymmetrical flap failure during approach to landing. After assessing the remaining controllability of the aircraft the Pilot performed a successful approach and landing.

Narrative

I was flying Aircraft X Cessna C-172N from ZZZ to ZZZ1. This was a personal flight to help maintain pilot proficiency. This is a rental aircraft from a flight school located at ZZZ. I did a full checklist-based pre-flight inspection; and normal takeoff from ZZZ. Flying my planned ZZZ1 RNAV XX approach; I proceeded to ZZZZZ; via ZZZZZ1 and ZZZZZ2 altitudes; and performed pre-landing checks. Had 4 white lights on the Runway XX PAPI; reduced power to lose altitude and speed. PAPI showed 2 Red 2 White; I adjusted power and attitude to maintain glide slope. Checked airspeed was in the white; added first notch of flaps. Waited for aircraft to stabilize; verified PAPI showed 2 Red 2 White; checked airspeed; adjusted power; added second notch of flaps. Almost immediately after second notch of flaps there was a loud bang; an immediate turn to the right; losing altitude very quickly. I corrected the uncommanded turn as much as possible; adjusted power to maintain altitude; looked for cause of problem. I noted one flap up; one down; and went around. Gained altitude; circled; and carefully explored controllability of the aircraft; 'no change' to the flaps was best for aircraft controllability. I landed without incident. In addition to the control system failure; the aircraft apparently suffered damage to the flaps. The flight school Mechanic; and the flight school; are not communicating with me; so I do not know the extent of the damage. A careful look at GPS data shows the aircraft was operated within limits; including Vfe; at all times prior to the incident. While this flight ended with a safe landing; for me this was a very close call; and caused me a few sleepless nights processing the event. I was mentally processing the event through the evening of Date1; and did not report the control system failure as quickly as I should have. It took me a while to overcome the seriousness of the event before I started my reporting responsibilities.Not having any communication or support from the flight school added additional delay. I was hoping for more information and cooperation from the school in the reporting process; as well as resolving what happened and how to avoid this situation in the future. I regret my delay and have reviewed my reporting duties and responsibilities as pilot in command to help ensure prompt reporting as required by regulations.