A B737 CREW REPORTS THAT AFTER DECLARING AN EMER AND DIVERTING TO A FOREIGN ARPT THEY DISCOVERED THEY MISAPPLIED AN ENG COWL ANTI-ICE VALVE CHECKLIST.

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THIS WAS DAY 3 OF A 19:19 SCHEDULED TRIP. DUE TO MECHANICAL DELAYS ON DAY 2 WE ARRIVED IN ZZZ2 AT XA27 LCL; APPROX 2 HRS LATE. WE FLEW 7:43 AND WERE SCHEDULED FOR OVER 7 HRS ON DAY OF EVENT. WE ARRIVED IN OUR HOTEL ROOMS APPROX XB30 AM. SLEPT FITFULLY DUE TO ARRIVING ACFT; BARKING DOGS AND NOISY HOUSEKEEPERS AND WERE AWAKENED REPEATEDLY BTWN XG00 AM AND XI00 AM. IN CRUISE AT FL400 ENRTE FROM ZZZZ1-ZZZZ2 DURING A DARK NIGHT; THE #2 ENG COWL VALVE OPEN LIGHT ILLUMINATED WITH THE RESPECTED SWITCH IN THE OFF POS. THE CAPT PULLED THE QRH WHILE THE FO SIMULTANEOUSLY ACARS DISPATCH TO COME UP ON FREQ. DURING THE DIAGNOSIS OF THE PROB; WE EVALUATED THE ENG COWL VALVE OPEN/TAI INDICATION AND THE ENG COWL ANTI-ICE CHKLIST TITLES. SINCE THERE WAS NO TAI INDICATION ON THE INST PANEL ASSOCIATED WITH THE LIGHT ON THE OVERHEAD WE THOUGHT WE MAY HAVE A BLEED AIR LEAK OR DUCT OVERPRESSURE AND USED THE ENG COWL ANTI-ICE CHKLIST. THIS CHKLIST SAYS TO DISENGAGE THE AUTOTHROTTLE AND RETARD THE THRUST LEVER UNTIL THE COWL ANTI-ICE LIGHT EXTINGUISHES. WE RECEIVED CLRNC TO DSND; BEGAN A DSCNT; AND FOLLOWED THE ENG COWL ANTI-ICE CHKLIST. WHILE DSNDING; WE CHKED COMMERCIAL RADIO AND FOUND A MESSAGE THAT DISPATCH COULD NOT CALL US AND FOR US TO CALL THEM. WE MADE 2 ATTEMPTS TO CALL DISPATCH; BUT BOTH CALLS FAILED. EACH CALLING ATTEMPT TOOK IN EXCESS OF A MIN. WE LOOKED UP COMMERCIAL RADIO AND QUICKLY REACHED THEM. THEY PATCHED US TO DISPATCH (SEVERAL MINS). CAPT EXPLAINED SITUATION TO DISPATCH/MAINT. INFORMED THEM WE REDUCED PWR ON AN ENG AND QUERIED THEM AS TO THEIR TAKE ON THE SITUATION. DISPATCH SAID THEY WOULD GET MAINT ON THE LINE. FOR THE NEXT SEVERAL MINS WE HEARD BACKGND NOISE. WE HEARD THE DISPATCHER ASK FOR MAINT TO SPEAK WITH US. WE HEARD MAINT ASK HOW TO USE THE RADIO. DURING THIS TIME THEY WOULD PERIODICALLY CALL US; BUT COULD NOT HEAR OUR REPLIES. WE SENT 2 COMMERCIAL RADIO MESSAGES. ONE TELLING THEM WE COULD HEAR THEM AND THE SECOND TELLING THEM ABOUT A POSSIBLE HOT MIKE. WE THEN HEARD DISPATCH CALL COMMERCIAL RADIO AND ASK IF WE WERE STILL THERE. COMMERCIAL RADIO ASKED FOR A SELCAL NUMBER AND THEN WE WERE SELCALED. HOWEVER; WE COULD NOT ESTABLISH COMS. AFTER A WHILE THE RADIO WAS SILENT. WE ATTEMPTED TO REESTABLISH WITH COMMERCIAL RADIO BUT WERE UNSUCCESSFUL. WE ATTEMPTED TO CALL VIA DTMF 2 ADDITIONAL TIMES; BUT WERE UNSUCCESSFUL. AT THIS POINT WE WERE UNABLE TO CONTACT THE COMPANY. WHAT WE BELIEVED TO BE THE COWL ANTI-ICE LIGHT (ENG COWL VALVE OPEN LIGHT) HAD NOT EXTINGUISHED PER STEP 2 OF THE COWL ANTI-ICE LIGHT CHKLIST. AT THAT MOMENT WE DISCUSSED OUR SITUATION AND AGREED WE HAD A BLEED AIR LEAK/OVERPRESSURE CONDITION. THE CHKLIST ENDS WITHOUT GUIDANCE WHETHER TO CONTINUE OR TERMINATE THE FLT. WE FELT THE SAFEST COURSE OF ACTION WAS TO DIVERT TO THE NEAREST SUITABLE ARPT. (WE WERE APPROX 100 MI N OF VOR. WE WERE ENTERING AN AREA OF SEVERAL HUNDRED MI THAT DID NOT HAVE A SUITABLE ARPT. WE DISCOUNTED ZZZZ3 AND ZZZZ4 DUE TO DISTANCE AND HAVING TO TRAVEL OVER WATER TO GET THERE. WE FELT ZZZZ5 WAS THE BEST CHOICE.) WE DECLARED AN EMER WITH CTR AND STATED OUR INTENTIONS TO DIVERT TO ZZZZ5. WE RECEIVED A CLRNC; PROGRAMMED THE RTE; AND COMMERCIAL RADIO OUR INTENTIONS TO DIVERT TO THE COMPANY. CAPT BRIEFED THE FLT ATTENDANTS AND PAX. WE CHKED PERFORMANCE DATA. WE REVIEWED THE 1 ENG INOP DSCNT; APCH; AND LNDG CHKLIST. WE APPLIED THE APPROPRIATE ELEMENTS SINCE BOTH ENGS CONTINUED TO OPERATE ALTHOUGH ONE AT IDLE THRUST. WE BRIEFED AND REVIEWED THE CHARTS. NEITHER ONE OF US HAD EVER BEEN TO ZZZZ5. IT WAS A DARK NIGHT WITH HIGH TERRAIN CONSIDERATION. WE ELECTED TO FLY THE VOR/DME APCH. APCHING VOR WE WENT TO COMMERCIAL RADIO TO SEND A MESSAGE OF OUR ETA. WE OBSERVED A MESSAGE ASKING US TO COME UP ON FREQ. ABOUT THE SAME TIME WE HEARD DISPATCH TRYING TO CALL US ON FREQ. THE DISPATCHER ASKED IF WE WOULD CONTINUE TO ZZZZ1 BECAUSE THEY HAVE MAINT THERE. WE INFORMED THEM WE WERE PREPARING TO LAND IN ZZZZ5. THE ACFT WAS CONFIGURED TO LAND AT FLAPS 15 DEGS MAX BRAKE. APCH AND LNDG WERE UNEVENTFUL. AFTER LNDG WHEN SPEAKING WITH THE FODO; WE DISCOVERED WE HAD ACCOMPLISHED THE ENG COWL ANTI-ICE CHKLIST IN ERROR. SEVERAL HRS PASSED. MAINT ARRIVED AND APPLIED MEL XX-X. WE THEN DEPARTED TO ZZZ2 AND ARRIVED UNEVENTFULLY. WE SHOULD HAVE TAKEN MORE TIME TO CAREFULLY DIAGNOSE AND DISCUSS THE SITUATION BEFORE DETERMINING A COURSE OF ACTION. THE LACK OF AVAILABLE SUPPORT DUE TO COM PROBS VIA COMPANY RADIO AND COMMERCIAL RADIO HAMPERED OUR ATTEMPTS TO PROPERLY EVAL AND MANAGE THE PROB WHILE OPERATING THE ACFT SAFELY. THE RESOURCES WE ARE ASSURED EXIST PER THE FOM WERE NOT AVAILABLE TO US. OUR FOCUS WAS ON RESOLVING THE SITUATION AND THEN TERMINATING THE FLT SUCCESSFULLY. OUR WORKLOAD WAS VERY HIGH IN PART DUE TO THE INABILITY TO COMMUNICATE. IMPROPER DIAGNOSIS OF MECHANICAL ISSUE.