PA-28 Flight Instructor reported experiencing loss of engine power during takeoff resulting in rejected takeoff

Narrative

The plane preflight looked good. Fuel tanks were full and oil was nice and clear and at 6.5. Taxiing the plane ran fine. We picked up our CRAFT (Clearance Limit; Route; Altitude; Frequency; Transponder code) clearance for IFR flight. Student completed run up and mag check looked good. 100' rpm drop on each mag. Max difference under 50'. Carb heat check provided slight drop. Everything was normal. ATC had us hold short for roughly 7-8 minutes and roughly 6 of those minutes of holding short we were behind a Socata TBM aircraft. We were roughly 4-5' behind the aircraft. We spoke of the exhaust smell coming from the plane in front of us and had our door open; mixture leaned; and rpms at 1200. I noticed oil temperature was a little above 180 and told student to increase RPMs to 1400 to increase oil pressure and engine cooling. ATC told us we were cleared for takeoff XXR and flying heading XXX. Student added mixture full rich; confirmed all overhead switches are on except for nav light; confirmed heels were on the ground and rolled onto the runway for takeoff. When adding max power; the plane proceeded to gain airspeed. We confirmed 3 in the green and airspeed was alive. However around 50ish kts; the aircraft wasn't increasing in power. Rpms were showing roughly 2200 rpms. Density altitude was 2930. Student had his hand on the throttle to ensure max power and had it on the bottom to confirm throttle is max position. I put my hand on the throttle to prepare for rejected takeoff as the speed at which the aircraft was accelerating was decreasing and roughly 59kts student started rotating at between 1 and 2 for XXR which is roughly 1500-1600' with roughly 2000' runway remaining and not enough power. The aircraft made it roughly 5' off the ground before I pulled the throttle; and continued to slow the aircraft. The aircraft bounced maybe once as it tried to slow then our brake effectiveness kicked in and I maneuvered the aircraft slightly to the left of centerline roughly 5' from the departure end of the runway without hitting the runway lights and no damage to the aircraft. We landed roughly 5' beyond the departure end of the runway. The engine failure during takeoff roll items were completed except for battery master as we communicated to ATC. Both souls onboard were okay. To improve this scenario; I would've rejected the takeoff sooner especially given the shorter runway distance.