C172 Flight Instructor reported a flap failure in the down position after attempting to clean up the aircraft after a slow flight maneuver. The instructor landed the aircraft successfully.

Narrative

Full scale split flap deflection at 3000 ft. resulted in an uneventful landing at ZZZ. I did not request assistance; but I should have and will next time. I believe I may have violated regulation by taking a student pilot on the flight; next time I will verify the test flight is in the aircraft log; this was a serious oversight on my part. Flap failure occurred during recovery from slow flight in a 200FPM climb at 80kts; 2500RPM; 3000MSL. We heard a loud bang; behind and above us; also to the left a bit; The airplane rolled right and I corrected by holding nearly 80% left aileron; and 40% left rudder. We lost 500 ft. during the recovery and turned about 50 degrees to the right. I pulled power; fearing too much asymmetrical lift but we were unable to maintain altitude; added power; 2400rpm and countered increased roll with additional rudder and aileron; we were able to maintain 75-80 kias and level flight at 2500MSL with 60% left rudder and 95% left aileron inputs. To turn left I could get about a 5 degree bank with full left aileron and additional rudder; to turn right I took out some left aileron and rudder input. Aircraft created additional drag and required a shallower AOA to maintain airspeed. We got the WX at ZZZ; decided RWY XX based on winds from 060. Although the common practice is to land with cross wind from the flap up side (the direction the aircraft wants to roll) I was worried a cross wind of 60 degrees could have made our approach unstable. During the approach I kept the airspeed around 75kts; using nose down trim and reducing power. I did not round out or flare normally; we landed on the upwind wheel and I added right rudder just after touch down; hoping to avoid side loading; we landed just to the right of center line. I determined I would not attempt a go-around; unsure of climb characteristics and positive that a landing in the grass was better than stalling on climb out; with the asymmetrical lift and the additional drag; I was very cautious of changing aircraft configuration to climb. ZZZ tower offered assistance; I declined; but should have accepted. In hindsight; although the approach felt fairly stable; the uneven nature of the flight controls could have resulted in unusual flight characteristics; including rolling or flipping during landing. Improper repair after identical failure 4 days prior. This also could have been prevented by a proper inspection of the flap track guide during the 100 hour. The worn flap track appears to have damaged the bolt that rolls back and forth; causing the flap to jam. I needed to test the minimum controllable airspeed to establish a safe approach speed at altitude. Should have tested flight characteristics at various airspeed; power settings and flight attitudes.I should have requested [priority] assistance. Should have verified test flight was completed by checking with maintenance directly and eliminating the middle man. To land on the shorter runway was helpful and led to a stable approach with headwind instead of a cross wind. I chose not to cycle the flaps - the loud noise indicated a snapped cable and I stand that it was correct to leave the flaps as they were; cycling the flaps could have led to worse asymmetry or unstable conditions - maybe negative flaps?