EA32 CREW TURNED ONTO A CLOSED TXWY.

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UPON ARR AT LAX AFTER A 4 HR SERIES OF DELAYS THAT PUT US IN AT XD50 AS OPPOSED TO OUR XA45 SCHEDULED ARR TIME; WE FOUND THE WX TO BE CAT III FOR OUR ACFT. TOUCHDOWN RVR WAS 1000 FT; MID 1200 FT; ROLLOUT 800 FT. WE BRIEFED FOR AN AUTOLAND; WHICH IS SOP FOR THE A320 WHEN WX IS CAT II OR BELOW. DURING OUR BRIEFING; WE NOTED THAT THE FOLLOWING TURNOFFS AND TXWYS WERE NOTAM'ED CLOSED BTWN RWY 25L AND ONE OR BOTH OF THE PARALLEL TXWYS BTWN RWY 25R AND THE TERMINAL COMPLEX; THUS ELIMINATING THEM AS POSSIBLE RWY EXIT PLANS. THESE TXWYS WERE J; K; N; AND P. TURNOFF TXWY M WAS THEREFORE PLANNED FOR OUR EXIT. WE PERFORMED AN UNEVENTFUL AUTOLAND WITH THE MAX LNDG AUTOBRAKE SETTING; VISIBILITY WAS AS ADVERTISED. THE CAPT DISCONNECTED THE AUTOPLT AT BTWN 100-80 KTS AS OPPOSED TO 30 KTS. MY ATTN WAS THEREFORE DIVIDED BTWN LOOKING TO MY R FOR OUR EXIT AND BACKING UP THE CAPT AND MAKING SURE WE TRACKED THE CTRLINE OF THE RWY. AS WE DECELERATED; I SPOTTED THE TXWY 'K' SIGN AND MADE A MENTAL NOTE THAT OUR NEXT TXWY WOULD BE TXWY M. I REDIRECTED MY ATTN OUT FRONT AS WE WERE STILL DOING 40 KTS OR SO. WE BOTH THEN SPOTTED A SET OF GREEN RWY TURNOFF LIGHTS. I ESTIMATED THAT ENOUGH TIME HAD ELAPSED AND THAT THIS MUST BE TXWY M. I THEN SAW THE TXWY M SIGN BUT FAILED TO RECOGNIZE THAT IT WAS PLACED AFTER THE LIT TURNOFF; NOT BEFORE. AS WE MADE THE TURNOFF; WE REALIZED THAT WE HAD MISTAKENLY EXITED ON TXWY K INSTEAD OF TXWY M. WE WERE THEN ISSUED A 'STOP' ORDER FROM THE TWR. THE TWR THEN QUERIED US IF WE SAW CONSTRUCTION VEHICLES AHEAD; TO WHICH WE REPLIED IN THE NEGATIVE. ALL THE CONSTRUCTION WAS BEING CONDUCTED TO THE E (BEHIND) US ON OTHER TXWYS; AND TXWY K WAS CLR AND UNOBSTRUCTED. OUR POS WAS THEN VERIFIED BY A 'FOLLOW ME' VEHICLE. THE TWR DISPLAYED SOME CONFUSION ABOUT THE LOCATION OF THE FOLLOW ME VEHICLE AND US. THE TWR THEN CLRED ANOTHER ACFT FOR TKOF ON THE RWY WE JUST LANDED ON (RWY 25L). EVEN THOUGH THE FOLLOW ME VEHICLE VERIFIED WE WERE CLR OF THE RWY; NEITHER MY CAPT NOR I FELT COMFORTABLE THAT THERE WAS SUFFICIENT DISTANCE BTWN US AND THE RWY AS WE COULD SEE THE FIRST SET OF HOLD SHORT LINES ABOUT 15 YARDS AHEAD OF US. THEREFORE WE TAXIED FORWARD FOR OUR OWN PIECE OF MIND. THIS WAS NECESSARY SINCE TWR NEVER SAID ANYTHING TO US AFTER TELLING US TO STOP. TWR THEN CLRED US TO CROSS RWY 25R AND CONTACT GND. SO WE TAXIED ON TXWY K ACROSS; THEN CONTACTED GND FOR TAXI TO OUR GATE. NO FURTHER COMMENT WAS MADE BY TWR OR GND. THERE WERE NEVER ANY VEHICLES ON TXWY K OR ITS IMMEDIATE VICINITY. CONTRIBUTING FACTORS TO THIS CONFUSED EVENT WERE: LACK OF SMGS GUIDANCE AT LAX. HAVING LEADOFF LIGHTS FOR A TXWY NOTAM'ED CLOSED 'ON.' FATIGUE ASSOCIATED WITH FLYING AN ALL-NIGHTER AS OPPOSED TO A LATE NIGHTER. SUGGESTIONS: ONLY HAVE LEAD-OFF LIGHTS ON FOR TXWYS AND TURNOFFS THAT ARE TO BE USED AND NOT ON FOR CLOSED TXWYS AND TURNOFFS.