I think 3 weeks is far too long to spend in Korea over xmas, unless you have family or friends here.
The weather will be cold, which takes out the option of Jeju or beaches etc. Busan is a nice place, but again in winter I dont know how nice.
Seoul will be cool place to be any time of year, but only for a few days (2/4max)
Air travel is not cheap out of here, especially on Asiana and Korean. There are some good deals to Taiwan as there is much competition on the route. Another option would be to use Cebu Pacific for a mid-priced flight to the Philippines? You might get good deal to HK also.