SR20 flight instructor on training flight with student reported a loss of aircraft control during landing rollout.

Narrative

Aircraft X came within 7 to 10 ft. of hitting multiple runway edge lights and a runway excursion during a high-speed landing rollout. The approach flown by the student; Pilot Flying; to Runway XX was relatively stable and the '300 ft.; stable' callout was completed. The instructor; Pilot Monitoring; who was a substitute for the primary IP and had never flown with the student before; was assessing the student on normal approach and landing for the syllabus-defined completion standard 'with minimal assistance.' The landing touched down hard and left of centerline on the left main gear and very nose-high with 5 percent to 10 percent power. The Pilot Flying corrected back to centerline with rudder but still 5 percent to 10 percent power in.The Pilot Flying; seeing the high-speed taxiway 1; decided to exit the runway there and began making rudder/braking inputs to get there. However; as the turn began; the SR20 was still above 40 KTS Ground Speed and thus the aircraft was unable to make the turn off at 1. The aircraft kept rolling but began to deviate left of centerline. The Pilot Monitoring (instructor) took the flight controls and firmly applied the toe brakes with as much force perceived to just avoid locking up and skidding. The aircraft came to a complete stop with the runway edge lights 7 to 10 ft. in front of it. The instructor (now Pilot Flying) taxied the aircraft off Runway XX and contacted ground per ATC's instructions. It is unclear when the power was reduced to idle during the event. In the FOM; expected taxi speeds exist but no such speeds exist for landing rollouts. Given the non-steerable nose wheel of the Cirrus SR20; a maximum runway exit speed could be implemented to give pilots a ground speed to look for that will ensure adequate directional control. Instructors could also be trained on this hazard (turns during landing rollout) during IP standardization. [I suggest] for manufacturers [to] publish a recommended runway exit speed in the POH/AFM to guide pilots. For FAA: include landing rollout in risk management or skills elements of the ACS (Airman Certification Standards) for all approach and landing tasks.