INSTRUCTOR WITH STUDENT DEPARTS WRONG RWY AT BTL.

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I WAS INSTRUCTING A PVT PLT STUDENT IN A COMPLEX AIRPLANE (C-172RG). AFTER COMPLETING TRAINING MANEUVERS; WE FLEW TO BTL FOR STOP AND GOES. WE WERE USING RWY 31 L TFC. OUR FIRST APCH WAS HIGH AND ENDED WITH A GAR. OUR SECOND APCH; WE LANDED LONG; AND WERE UNABLE TO TKOF BECAUSE OF A LACK OF RWY IN FRONT OF US. WE TOLD THE TWR THAT WE NEEDED TO TAXI BACK. THEY TOLD US TO DO A 180 DEG TURN ON THE RWY AND HOLD SHORT OF RWY 5; WHICH WE DID. A KING AIR DEPARTED RWY 5. THE TWR TOLD US THAT WE COULD TKOF ON RWY 5 FROM THE INTXN OF THE RWYS; AND THAT 2700 FT WOULD BE AVAILABLE FOR TKOF. FROM MY EXPERIENCE; THE AIRPLANE WOULD NEED ABOUT 1700 FT FOR TKOF. SO; I CONCLUDED THAT WE COULD DEPART FROM THE INTXN. WHAT I FAILED TO REALIZE WAS THAT WE WERE ON RWY 31. WE TOOK OFF FROM THAT RWY. AFTER TKOF; THE TWR INFORMED US THAT WE TOOK OFF FROM THE WRONG RWY. WE FLEW BACK TO OUR HOME BASE WITHOUT INCIDENT. I BELIEVE THIS INCIDENT OCCURRED FROM LACK OF SLEEP CLOUDING MY JUDGEMENT. THE NIGHT BEFORE I FLEW ON A 4 HR IFR XCOUNTRY. WE LANDED AFTER MIDNIGHT (LCL TIME). I AWOKE EARLY (LCL TIME) THAT MORNING. I FLEW TWICE THAT MORNING FOR ABOUT 2 HRS; AND ONCE THAT AFTERNOON. I FELT TIRED; BUT HAD NO IDEA HOW BAD MY JUDGEMENT HAD BECOME. IN THE FUTURE; I WILL NOT SCHEDULE FLTS FOR LATE AT NIGHT AND EARLY THE NEXT MORNING. I REALIZE THAT THE 67 HR MINIMUM MAY NOT BE ENOUGH AFTER A TOUGH; LONG FLT.