I flew LHR-SYD vv. in J class on Asiana (OZ) in 2008 and fine flights they were.
Same problem with OZ, on the return leg you need an overnight, they do pay for it but at a hotel some distance from ICN, so I paid own dime to stay in the Hyatt Incheon.
At the time both OZ and KE were offering the free o/night. When I was checking into the Hyatt there were a load of KE J class pax being checked in for their overnight on KE vouchers, with KE ground staff assisting.
Not sure if that hotel is SOP, but it certainly looked like a well practised process.