Been in a similar boat - also NYC based and Korean American so travel quite frequently to US-ICN.
- Delta: There is Partner availability for KE at one way from JFK-ICN on Business at 115K but its EXTREMELY RARE. If I redeem Skymiles I will go via D1 via ATL/DTW/MSP just because redemption flights give me MQM but I would always pick KE>DL
- Asiana: Before the merger there is still availability from JFK-ICN via Asiana through S* partners Air Canada/Singapore Airlines for which I found one way on Aeroplan to be around 85K
- AA: AA has 1 daily ICN bound flight out of DFW and this could be a possibility via BA Avios as well or via Ethiad Program