Have a look at doing the type-rating with Air New Zealand in Auckland. Competitive prices, good training, and a 3 week course. Maybe a little more expensive and longer than in the USA, but if you can transfer the type-rating across to your JAA licence ok it will save the hassle of getting an FAA licence followed by a lengthy wait for them to process the paperwork.
Then there's the question if you REALLY want to join KAL as a B777 FO with no experience on type (relevant to your line training where many Captains get chopped), and no chance for the left seat ever?