LR35 WITH ZKC EXPERIENCED TFC CONFLICT AT FL380 DURING WX DEV.

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WE WERE IN LEVEL CRUISE AT FL380 (WRONG WAY) EBOUND. THIS WAS ASSIGNED BY A PREVIOUS CTLR FOR TFC. APCHING A LARGE CLOUD FORMATION; THE CAPT REQUESTED ATC CLRNC TO DEVIATE 10-15 DEGS AROUND THE STORM TO THE R; CLRNC WAS ISSUED; AND WE BEGAN A TURN OF ABOUT 5 DEGS TO TRY TO CLR THE SOUTHERN BOUNDARY. SOON AFTER THIS TURN WE DETERMINED THAT ANOTHER 10 DEGS WOULD BE NEEDED; AND THE CAPT TUNED THE HDG BUG ANOTHER 10 DEGS TO THE R. AT THIS POINT WE ENTERED IMC CONDITIONS AND EXPERIENCED SOME MODERATE TURB FOR A BRIEF TIME. UPON CLRING THE CELL ON RADAR WE TURNED BACK TO A DIRECT COURSE TO FAM VOR; AND ATTEMPTED TO ADVISE ATC. THE FIRST XMISSION WAS BLOCKED OR UNHEARD. THE RESPONSE TO MY SECOND XMISSION WAS 'TURN L HDG 100; AND DSND TO FL370; EXPEDITE DSCNT.' GIVEN THAT WE WERE ALREADY FLYING HDG 095 DEGS; A L TO 100 DEGS SEEMED UNUSUAL; FIGURING A LAG IN RADAR RPTING I ADVISE ATC THAT WE WERE ALREADY DIRECT TO FAM; HDG 095 DEGS; BUT WE WOULD 'QUICK DSND TO 370.' THE NEXT CALL ON THE RADIO WAS A 30 DEG COURSE CHANGE FOR AN ACR FOR TFC AT FL380. SHORTLY AFTER THIS CALL OUR ACFT EMERGED FROM IMC CONDITIONS; AND I OBSERVED A LARGE ACFT AT MY 2-3 O'CLOCK POS AT ABOUT 5 MI. I ADVISED ATC THAT IF IT HELPED I HAD AN ACFT IN SIGHT; AND THAT HE WAS CLR OF OUR POS. THE CTLR ACKNOWLEDGED WITH A THANK-YOU. COMPOUNDING CIRCUMSTANCES: 1) ACFT ASSIGNED WRONG WAY ALT; AND THEN REMAINED THERE. NEITHER THE CREW OR ATC CORRECTED THIS AFTER THE INITIAL PURPOSE WAS FULFILLED; THOUGH EITHER COULD HAVE. 2) ACFT CLRED FOR A WX DEV INTO THE PATH OF APCHING TFC; AT THE SAME ALT. 3) FLT CREW DELAY IN REQUESTING DEV; CAUSING FLT INTO POOR CONDITIONS; THUS CREATING HIGHER WORKLOAD; AND THE INABILITY TO SEE AND AVOID. 4) FLT CREW'S MISJUDGMENT OF INITIAL HDG TO CLR CLOUD; AND ASSOCIATED HAZARDS. 5) ATC FAILURE TO PROVIDE FLT CREW WITH SITUATIONAL INFO ABOUT TFC; AND ALERT CREW TO THE GRAVITY OF THE CURRENT SITUATION. 6) ACFT LACK OF TFC MONITORING EQUIP TO ALERT FLT CREW OF CONFLICTING TFC. 7) FLT CREW AND ATC LACK OF EFFECTIVE COM LEADING TO MISSED RADIO CALLS; AND MISUNDERSTOOD RADIO CALLS. BASED ON MY OBSERVATION; I HAD NO IDEA THERE WAS EVEN A CONFLICT OF A REPORTABLE NATURE UNTIL ADVISED BY GND CTL TO CALL KANSAS CITY UPON LNDG; ALMOST AN HR LATER. BOTH ACFT FLT CREWS; AND THE ATC CTLR WERE SURPRISINGLY CALM DURING THE CONFLICT. THERE WAS NO URGENCY OR PANIC IN ANY OF THE XMISSIONS; THOUGH PERHAPS IF THERE WERE; BETTER AWARENESS OF THE SITUATION WOULD HAVE BEEN REALIZED.