AFTER LNDG AN MU3 FLC EXPERIENCED BRAKING PROBS. WHEN THE ACFT FAILED TO DECELERATE; THE FLC PERFORMED A GAR AND DIVERTED TO THEIR ALTERNATE. A PRECAUTIONARY GEAR EXTENSION WAS MADE ENRTE AFTER APCH CTL STATED THAT THE ACFT MAY HAVE CLIPPED A LIGHT AT THE OPPOSITE END OF THE RWY DURING THE GAR. DURING LNDG ROLLOUT AT THE ALTERNATE ARPT; THE L BRAKE FAILED AND THE L HEAT PLUGS DEFLATED THE TIRE.

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DSNDING OUT OF 12000 FT WITH ANTI-ICE ON WE BEGAN TO ENCOUNTER MODERATE ICING IN THE CLOUDS. WE WERE GIVEN VECTORS FOR THE APCH TO THE ILS RWY 3 TO MKC CIRCLE TO LAND ON RWY 1. AFTER PICKING UP THE ARPT WE ELECTED TO CIRCLE TO LAND ON RWY 1 AT CIRCLING MINIMUMS OF 1400 FT. THIS APCH WAS LIMITED TO A FLAPS 10 DEG LNDG DUE TO THE ICING CONDITIONS; SO OUR SPD WAS REF +10 KTS. A RPT OF FAIR TO GOOD WAS GIVEN TO US ON RWY CONDITION BY AN ARPT VEHICLE. WITH TURB ON FINAL APCH WE WERE REF +20 KTS. UPON TOUCHDOWN AT APPROX 1500 FT OUT OF 7000 FT WE APPLIED THE BRAKES. AFTER A BRIEF SENSATION OF BRAKING THE ACFT WAS NOT DECELERATING. WE ELECTED TO GAR. ON CLBOUT APCH ALERTED US THAT WE MAY HAVE CLIPPED AN APCH LIGHT ON THE OPPOSITE END OF RWY 1. SO WE EXTENDED THE GEAR AND 3 GREEN APPEARED. OUR ALTERNATE WAS MCI. ON APCH INTO MCI WE WERE PICKING UP MODERATE ICING AND TURB. WE TOUCHED DOWN ON THE 1000 FT MARKERS AT MCI. UPON TOUCHDOWN AFTER THE BRAKES WERE APPLIED THE L MAIN TIRE BLEW. WE MAINTAINED ACFT CTL AND EXITED THE RWY 1R AT THE END. NO INJURIES. JUST A FAILED L BRAKE AND A BLOWN L TIRE. SUPPLEMENTAL INFO FROM ACN 327015: HEAT PLUGS LET L TIRE DEFLATE.