A B737 FLC RECEIVED A #2 ENG FIRE WARNING AND WERE UNABLE TO EXTINGUISH THAT WARNING THROUGH THE USE OF EMER PROCS. THE FLC DID NOT DECLARE AN EMER; BUT DID INFORM ATC OF THEIR SIT. WHILE IN A MODIFIED TFC PATTERN ON A FAST TO LNDG; THE CAPT OVERSHOT THE FINAL APCH CTRLINE WITH NEARBY TFC ON A PARALLEL APCH.

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ON ARR TO ICT WE WERE JUST HANDED OFF TO TWR. #2 ENG FIRE WARNING. WHILE FO (PNF) RAN CHKLIST AND SECURED ENG; I NOTIFIED TWR WE'D BE LNDG SINGLE-ENG AND REQUESTED EQUIP TO BE STANDING BY. DIDN'T DECLARE EMER; BUT STATED WE HAD A FIRE WARNING PROMPTING THE SHUTDOWN. HAD TO FIRE BOTH BOTTLES AND FIRE WARNING HANDLE STILL ILLUMINATED. EXPEDITED TO RWY 1L AT FLAPS 15 DEGS; GEAR DOWN; AND 160 KIAS. AT THAT SPD AND RESTRICTING BANK TO 15 DEGS; IT WAS DIFFICULT TO KEEP OUR TURN FROM A MODIFIED DOWNWIND AND BASE TO FINAL INSIDE OF ANOTHER ACR'S DC9 TURNING R BASE TO FINAL FOR RWY 1R. I NOTIFIED TWR THAT MY TURN WOULD TAKE ME SLIGHTLY THROUGH CTRLINE AND THAT I HAD THE DC9 IN SIGHT AND SLIGHTLY AHEAD. THERE WAS NO CONFLICT; BUT IT WAS AN EXTRA DISTR I DIDN'T NEED RIGHT THEN. SINCE AS FAR AS WE KNEW WE STILL HAD A FIRE AND THE RWY WAS LONG; I WAS GOING FAST AND I FOUND JUDGING THE TFC SEPARATION TOOK MORE CONCENTRATION THAN EXPECTED. IN THE FUTURE I'LL NOT SOUND SO 'COOL' OVER THE RADIO; DECLARE AN EMER AND GET THE TWR TO MOVE THE TFC.