I only know one japanese drama sorry which is Boku no Ita Jikan.
It's ok that'love. He has a mental illness.
An upcoming drama (which IMO beats everything in the category): Remembering Li Chuan's Past. I personaly didn't like the novel, but I can guarantee illness is a serious issue in their story.
Not Korean but Japanese.

Kou Kou Kyoushi (2003) 
I presume you have seem
http://mydramalist.com/5136-i-will-fly-to-the-sky-... from japan.
as for korean, i recoomend
it's okay, that's love

Okay this is not Korean but;
http://mydramalist.com/790-kou-kou-kyoushi-2003 male lead has a deadly disease, teacher-student romance and more. Just watch it, it's a masterpiece.
Nirvana in Fire - Historical/political chinese drama, male lead is extremely weak with a sick body
I don't know any Korean ones  but I second "Koizora" for Japanese.  I know you're requesting for Korean but just in case, putting these out there: "Autumn's Concerto" and "The Way We Were" (oops spoiler alert!) which are both Taiwanese dramas. Loved these dramas so much so I can guarantee you that you won't regret watching them :D
It's Okay, That's Love
Fated to Love You
Aishiteru to Ikketure - he's not sick, just hearing impaired

Itoshi Kimi E & Someone Like You - again not sick, but has a disability for a portion of the drama

I can't remember if anyone mentioned Kill Me, Heal Me - mental illness

It's Okay, That's Love - mental illness
The Greatest Love (2011)
Kill Me, Heal Me
Good Doctor
Mask
Kill me heal me
It's okay that's love
Why why love - taiwanese
love beyond frontier 2 - Thai 
two spirits love - Thai 
Sorry, i love you - but it is super super sad and the ending still makes me cry.
fated to love you (korean) - He loses his memory (although its been a while since i've seen it)
it's okay, thats love - Kwang Soo's torrets acting is just amazing. and Ju In sung's mental issues are well portrayed too.
Mask - He loses his memory - Good drama (with ju ji hoon)
Modern Farmer - Lee Shi Eon character i think is sick during the drama (again it's been a while since ive seen it)
Dr Frost - Partial memory loss due to psychological trauma