FLT ASSIST RENDERED BY DAY TRACON TO A C152 WITH FLAPS LOCKED OUT AT THE 30 DEG POS AND A ROUGH ENG. EMER DECLARED BY INSTRUCTOR PLT.

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PRIMARY STUDENT AND I HAD BEEN PRACTICING SLOW FLT IN PREPARATION FOR HIS UPCOMING CHKRIDE. THE FLAPS HAD BEEN EXTENDED AND THE ACFT SLOWED WHILE AT 3000 FT MSL W-SW OF DAY. WHEN I ASKED THE STUDENT TO RETURN TO CRUISE FLT; HE DUTIFULLY PLACED THE FLAP SELECTOR IN THE ZERO DEG OR RETRACTED POS AFTER FULL THROTTLE WAS APPLIED. HOWEVER; AS THE STUDENT NOTICED AND EVENTUALLY FIGURED OUT; THE AIRSPD REMAINED SLOW BECAUSE THE FLAPS HAD NOT RETRACTED. AS WE LATER FOUND; A LIMIT SWITCH HAD FAILED AND EFFECTIVELY JAMMED THE ELECTRICALLY ACTIVATED FLAPS IN THE 30 DEG DOWN POS. THE FIRST THING WE DID WAS DETERMINE THAT ALT COULD INDEED BE MAINTAINED WITH AN AIRSPD OF APPROX 65 KTS AT APPROX 2400 RPM. WHILE SORTING OUT THE EXTENT OF OUR DIFFICULTY; THE FLT PROCEEDED SLOWLY NEBOUND. WE CHKED THE POH AND THE EMER CHKLISTS BUT FOUND NO PROC ADDRESSING OUR PARTICULAR PROB. BY THIS POINT WE WERE JUST W-SW OF VERONA AND DECIDED THAT WE DID NOT WANT TO DSND UNTIL WITHIN GLIDING DISTANCE OF OUR CHOSEN ARPT; 3I7; WHICH LIES BENEATH THE OUTER RING OF DAY CLASS C AIRSPACE. WE CALLED DAYTON APCH; EXPLAINED OUR FLAP SIT AND THE FACT THAT OUR ENG WAS RUNNING A BIT ON THE ROUGH SIDE; AND WE DECLARED AN EMER. ONCE RADAR-IDENTED; APCH ADVISED THAT 3I7 WAS APPROX 8 MI DISTANT AND I62 WAS APPROX 7 MI DISTANT. IN THE SAME BREATH; HE ANNOUNCED THAT THESE ARPTS WERE ESSENTIALLY EQUIDISTANT FROM OUR POS. OUR ENG ROUGHNESS HAD SUBSIDED HAVING APPARENTLY BEEN THE RESULT OF AN OVERLY RICH MIXTURE/'LOADING UP' DURING SLOW FLT THE PREVIOUS 1/2 HR OR SO. WITH A NOW SMOOTH RUNNING ENG WHOSE TEMP WAS SOLIDLY IN THE GREEN AND MAINTAINING ALT WITH CRUISE PWR ONLY; I ELECTED TO LAND AT 3I7. ATC PROVIDED VECTORS AND OFFERED TO CALL AHEAD TO NOTIFY THE ARPT AND HAVE ANY AVAILABLE CRASH EQUIP STANDING BY. THROUGHOUT THE EVENT; DAYTON APCH WAS VERY PROFESSIONAL/HELPFUL; AND I CALLED THEM AFTERWARD TO THANK THEM FOR THEIR HELP AND TO LET THEM KNOW WE HAD LANDED WITHOUT INCIDENT. I HOPE THAT MY STUDENT LEARNED HOW TO REACT TO AN UNUSUAL SIT; HOW TO ANALYZE THE CIRCUMSTANCES TO DETERMINE THE IMMEDIACY OF THE THREAT AND SEARCH THE PROPER SOURCES/CHKLISTS FOR A SOLUTION; THEN NOTIFY THOSE WHO CAN HELP IN CASE THEIR ASSISTANCE BECOMES NECESSARY TO REACH A PLACE OF SAFETY.