Is it possible to extend your stay, to at least 2 weeks? Some people have done another procedure the day after they did their eyelid surgery, but it depends on what you are doing to your eyes - ex: if you get epi, you wouldn't want your inner corners to get tugged on when the rhino doctor is fixing your nose. Often clinics will recommend at least waiting till the stitches for your eyes are out (usually 6-7 days post op), so you could do rhino the day after your stitches are out, and then you need to stay 5 to 7 days after rhino (Dr Kim said cast usually comes off in 7 days, but it can also be removed after 5 days if one is short on time).
If it's not possible to extend, then you can try to see if you can do rhino before the stitches in your eyes come out. Either way, you may have to book the doctors before you go to Korea.