A NON TCASII EQUIPPED LGT FREIGHTER HAS AN NMAC WITH A NORDO ACFT ON MISSED APCH FROM RWY 4R AT JFK AT NIGHT. RADAR AT N90 HAD FAILED TO DETECT THAT OTHER ACFT.

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LOW VISIBILITY APCH INTO JFK. UNABLE TO GET ESTABLISHED ON FINAL AND ELECTED TO GAR ON RWY 4R. ILS MISSED APCH CLB TO 500 FT; THEN CLBING R TURN ON A 100 DEG HDG. AS TURN WAS INITIATED APCH WAS NOTIFIED. WITHIN THE NEXT 30 SECONDS WE WERE CLRED TO CLB TO 4000 FT THEN LEVEL AT 3000 FT (ALREADY AT 3300 FT) THEN TURN L 360 DEGS; THEN TURN R TO 060 DEGS. AT 2000 FT CLBING; SPOTTED ACFT WITH WHITE WING LIGHTS 1/2 THE WIDTH OF MY WINGS. ALTERED HDG AND ALT TO AVOID. HARD TO TELL WHAT HE WAS DOING. ADVISED APCH WE HAD TFC IN SIGHT. APCH WAS NOT TALKING TO HIM. WE NEED TCASII IN ALL ACFT. DURING THE RAPID MANEUVERS; THE SO GOT VERTIGO REACHING FORWARD TO SET THE PWR. THE FO GOT FIXATED ON THE CLRNC TO INTERCEPT V1 TO LEEAH INTXN. HE SPENT A GOOD MIN TRYING TO FIND LEEAH. HAD TO STOP HIM AND GET V1 ESTABLISHED. WENT THROUGH V1 AND WAS GIVEN A VECTOR OF 180 DEGS TO WHITE INTXN. IT TOOK A GOOD 10 MINS TO GET A FUNCTIONING CREW. IT WAS CLOSE; VERY CLOSE. OTHER ACFT WAS NOT TALKING TO KENNEDY APCH. UNABLE TO DETERMINE HDG OR ALT OF OTHER ACFT DUE TO RAPID MANEUVERING.