Not quite sure where ICN came from in your latest post! But, in case it is merely because you wanted to see it and not for routing gymnastics, have you considered booking OW LHR-ICN? There is no direct flight so valid routings with stopovers include LHR-HKG-ICN-NRT-LHR using CX and JL as partners. You don't have the miles to do it all on mileage but you might be able to make spring for MFU's on BA sectors or something.

Alternatively, if you have the cash and you are willing to go to Korea... buy one-ways to Korea and then buy 2x OW RTW tickets, e.g. DONE3 (a three continent business class ticket - you could get economy (=LONE3) or premium economy (where offered) if business is stretching the budget) on the way back? Korea is currently one of the cheapest places to get one. You would get two further stop-overs in Asia (so HKG and NRT) and, if you go on from Asia to Europe, you would get to complete the RTW via Vancouver at a later date into the bargain! Look at the Oneworld site for current pricing but it's probably ~£1k for economy and ~£3k for business.

By the way, Korea is really quite alien(ating), more so than Japan: a ratio of 2 weeks Japan to 4 days Korea was enough for me. I took the sleeper train (possibly no longer running) from Tokyo to Shimnoseki, then the overnight ferry to Busan for a couple of days and then a day's trip by train to Seoul (via Geong-somewhere with royal palaces and tumuli) for a couple of days. I think there is now a TGV but the Busan-Seoul sightseeing was possible in a day without.