Flight Instructor reported experiencing an airborne conflict after takeoff with another aircraft executing a go-around when approaching the same runway.

Narrative

Conducting training flight in Aircraft X; I was the flight instructor sitting in the right seat. After runup and positioned at the hold-short line of Runway #1 at Lebanon Warren County airport; both student and instructor noticed Aircraft Y on the downwind and making calls as appropriate. When the Aircraft Y turned base; student pilot made the appropriate departure announcement; accelerated to the runway and set takeoff power. There was no delay with the process or stopping on the runway. During the takeoff roll; the approaching Aircraft Y announced on the CTAF that they were going around due to departing traffic. Both aircraft had the other in sight and there was never a collision danger. The event ended with the our departing to the south and the Aircraft Y remaining in the pattern. From the ground; both the instructor and the student estimated the Aircraft Y was 500+ feet AGL. Assuming a 500 FPM decent; this should have easily given us the needed time to safely depart from the runway. Unfortunately; not the case. The Aircraft Y closed extremely quickly. As the we departed the area; the instructor viewed the Aircraft Y's ADSB data and saw speeds in excess of 100 knots on short final during subsequent approaches to I68. This is far too fast for any part of the pattern and especially short final.