C172 EXPERIENCES NMAC WITH ANOTHER C172 IN TFC PATTERN AT TKI.

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ARPT WAS MODERATELY BUSY. MORE THAN 5; LESS THAN 10 OPS MONITORED DURING 10-15 MINS PRIOR TO APCH. OPS WERE USING RWY 35. CALLED IN 'E OVER THE LAKE; LNDG.' TWR'S RESPONSE WAS 'CLRED DIRECT TO RWY 17.' CALLED BACK FOR CONFIRMATION OF RWY 17. RESPONSE WAS; 'DID I SAY RWY 17? CLRED TO LAND RWY 35.' NO REQUEST FOR ANOTHER CALL-UP ON FINAL. DSNDED 2500 FT TO PATTERN ALT 1500 FT. PWRED BACK TO BEGIN BASE DSCNT. TWR CLRED A CESSNA FOR TOUCH-AND-GO; NO STATEMENT OF '#2.' AFTER TWR CLRED ACFT #2; I SCANNED TO THE R; LOOKING FOR ACFT #2. ACFT #2 APPEARED; OVERFLYING BY 100 FT; APPEARED TO BE MAKING BASE TURN FROM DOWNWIND. I BROKE TO S ABOUT SAME TIME TWR CALLED 'THERE ARE 2 OF YOU OUT THERE.' I CALLED THAT I HAD BROKEN OFF TO THE S. TWR RESPONDED TO THE EFFECT THAT 'I DIDN'T ANTICIPATE YOU ARRIVING SO SOON.' CONTINUED DSNDING TO ASSURE CLRNC FROM ACFT #2. BEGAN R TURN BACK TO BASE. ACFT #2 BELOW; TURNED TO FINAL. I ANNOUNCED I WAS DOING A 360 DEG TURN TO ACHIEVE SPACING. TWR APPROVED MANEUVER. CLBED BACK TO PATTERN ALT DURING CIRCLE. CALLED IN 'LONG FINAL' BEFORE PULLING PWR. TWR STATED I WAS 'STILL CLRED TO LAND.' NO CALLS FROM ACFT #2 THROUGH THE INCIDENT. ACFT #2 NOT IN SIGHT. NO TWR CLRNC FOR ANOTHER TOUCH-AND-GO DURING TAXI TO N END OF ARPT. THIS INCIDENT APPEARS TO BE CAUSED BY INATTN IN TWR -- 2 MISTAKES ON 1 APCH: CLRING ME TO RWY 17; THEN CLRING ACFT #2 FOR TOUCH-AND-GO WITHOUT SEQUENCING. TWR DID NOT NOTICE CONFLICT UNTIL I HAD ALREADY SEEN ACFT #2 APPEAR OVER MY R WING. I WAS NOT INSTRUCTED TO CALL IN ON BASE OR FINAL; BUT DOING SO PROBABLY WOULD HAVE ALERTED TWR BEFORE THE CONFLICTING CLRNCS WERE ISSUED.