Cessna Flight Instructor reported an NMAC with another light aircraft on approach to MHK airport.

Narrative

Student and I were performing right-hand traffic patterns for Runway 03 at MHK. ATC instructed us to follow a [light twin] on our way to land. Visual contact was maintained with the [light twin] until they did a full stop and departed the runway. As my student was turning base to final; we noticed an aircraft that ATC had not warned us about approximately 200-300 ft. away from us. I took control of the aircraft in order to perform evasive action. My first priority was to get away from the aircraft; then to re-enter the downwind when it was safe to do so. Specifically; the power was advanced to full; the aircraft was positioned in a way to climb away from the conflict aircraft; and flaps were retracted per the published go-around procedure.According to ATC transmissions; the conflict aircraft (I believe it was a low-wing complex aircraft) was performing a practice instrument approach and was supposed to report when they were passing the FAF; which they apparently did not. I am unsure if the conflict aircraft ever saw us.As far as I am aware; my student (Pilot Flying) and I (Pilot Monitoring) were in good physical and mental shape at the time of the incident; and I first noticed the conflict aircraft as I was visually scanning the final approach corridor as we were preparing to turn final. Initially; my student could not see the conflict aircraft due to blind spots inherent with the model of the aircraft.