A COMMERCIAL RATED PLT FLYING A PA28 NEAR CKB EXECUTED A MISSED APCH AFTER REALIZING A CONFLICT WITH ANOTHER ACFT CLRED TO LAND BEFORE HIM.

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I WAS UNDER THE HOOD IN L SEAT MAKING PRACTICE LOC 21 APCH TO CKB. TWR HAD ALREADY CLRED US TO LAND. AT ABOUT 8.6 DME FROM VOR (OR ABOUT 5 MI FROM RWY) TWR ASKED HOW FAR WE WERE FROM ARPT. I ANSWERED THAT WE WERE ABOUT 8 DME FROM THE VORTAC. TWR CLRED THE CESSNA TO ENTER PATTERN IN FRONT OF US. AT ABOUT 3 MI AND 2000 FT MY SAFETY PLT IN R SEAT SAW THE CESSNA BELOW AND TO OUR L ON BASE LEG. WE CALLED OUT TFC TO TWR AND ASKED WHAT THEY WISHED US TO DO. THE TWR DIDN'T RESPOND; SO WE CALLED AGAIN. FINALLY THE TWR TOLD US TO S-TURN AND LAND #2. I WAS TOO CLOSE AND FAST FOR THAT; SO I ASKED TO DO A 360 DEG TURN. TWR TOLD US TO DO A 360 DEG TURN TO R WHICH INVOLVED A LOT OF CLBING TO CLR HILLS NEAR ARPT. CONTRIBUTING FACTORS: BOTH LCL RADARS WERE OUT; AND IN SPITE OF '10 MI VISIBILITY' HAZE WAS VERY THICK; SO TWR COULDN'T SEE TFC IT WAS TRYING TO CTL. MY ANSWER TO TWR'S QUESTION WAS NOT WHAT SHE EXPECTED. SHE WANTED DISTANCE FROM ARPT; BUT I GAVE DISTANCE FROM VORTAC (MORE PRECISE; BUT NOT WHAT ASKED).