A Drama with sick male lead...

Falling for Innocence (he has heart's problems)

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.

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

Nirvana in Fire - Historical/political chinese drama, male lead is extremely weak with a sick body

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

Mask
Kill me heal me
It's okay that's love

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