C172 LANDS ON WRONG RWY AT FXE DURING NIGHT TRAINING FLT.

Narrative

I WAS TEACHING A STUDENT PLT NIGHT LNDGS. WE WERE ON A VISUAL APCH AND CLRED TO LAND ON RWY 7. WE WERE HAVING A DIFFICULT TIME FINDING/IDENTING THE ARPT BECAUSE OF ALL THE CITY LIGHTS. WE DIDN'T SEE THE RWY UNTIL WE WERE ONLY 2 MI OUT AND WELL ABOVE TPA. WE IMMEDIATELY BEGAN A QUICK DSCNT AND SET UP FOR LNDG. I WAS FOCUSING ON TEACHING THE STUDENT ABOUT NIGHT LNDGS -- LETTING THE STUDENT FLY THE AIRPLANE. WE ENTERED A SHORT L BASE; THEN FINAL; AND LANDED. BECAUSE WE GOT A LATE START ON THE APCH AND BEING DISTR BY INSTRUCTING A STUDENT; I DID NOT REALIZE WE WERE LNDG ON THE WRONG RWY. FORTUNATELY; IT WAS EARLY MORNING AND THERE WAS NO OTHER TFC. HAD THIS HAPPENED DURING A BUSY TIME; IT COULD HAVE BEEN A DANGEROUS SIT. THIS WAS DEFINITELY A WAKE-UP CALL FOR ME. I REALIZE THAT I AM ULTIMATELY RESPONSIBLE FOR WHAT HAPPENS IN THAT AIRPLANE AND THAT I MUST ALWAYS BE COMPLETELY AWARE OF WHERE I AM AND WHAT IS GOING ON. ANOTHER CONTRIBUTING FACTOR WAS THIS WAS A LONG XCOUNTRY TRAINING FLT; LATE AT NIGHT; AND WE WERE BOTH TIRED.