Instructor pilot reported a bounced landing resulted in a runway excursion on a training flight. The Instructor and Student evacuated the aircraft without injury and noticed the propeller damage.

Narrative

In the afternoon of Day 0; I became involved in an incident while providing flight instruction out of ZZZ. It was a routine training flight to the practice area. We began pattern work for runway XXR; and completed one lap in the pattern and one landing with no incident. On our second touch-and-go; the incident occurred. I was at the controls and had the power set to idle and flaps fully configured at +2. We floated down the runway about 100 ft. As I flared; the aircraft touched down and bounced once. I attempted to settle back down and the aircraft bounced again; higher. At this point; I applied full power to go around. However; the aircraft settled back down to the ground and bounced even harder this time. The aircraft was finally fully on the runway but I could not control it. I pulled power to idle and immediately applied full braking and full right rudder. I was unable to keep the aircraft on the runway and we careened left off the runway into the grass between the parallel runways. I finally stopped and shut down the aircraft and was able to view the propeller damage. Both blades were split and broken. After some reflection on this incident; I think there are a couple factors involved. First off; I should have applied full power for the go-around after the first bounce. The porpoising became much worse after that first bounce and I believe this could've helped to save the landing. Secondly; I should've held in some power to avoid the float and fast rate of descent that we initially experienced. This is something I do for most landings; but in this aircraft I've found that the aircraft will continue gliding down the runway with any power left in. Finally; I am aware that this exact aircraft experienced a similar incident with the previous owners. While I don't have all the details; I am aware that a student before me suffered a prop strike due to the aircraft porpoising down the runway.As I was PIC and providing instruction to a student pilot; I realize that all actions and inactions involved are my responsibility. I took the corrective action to go-around in the aircraft before the propellers broke and was unsuccessful; and am still confused as to why this is. I am happy to provide any further insight on this incident; and am open to comments and advice.