Two Weeks (2013)

Jang Tae San works at a pawn shop by day and prostitutes himself to older women at night. One day, he's contacted by his ex-girlfriend that he is the father of her daughter with leukemia, and he's the last chance to donate bone marrow and save their daughter's life. That same day, he finds himself framed for murder but manages to escape and runs away. With nobody to trust, Jang Tae San has two weeks to try to prove his innocence and survive until his daughter's surgery. Park Jae Gyeong is a prosecutor searching for evidence to take down Moon Il Seok, the man who stabbed her father, and Cho Seo Hui, a beloved senator who is actually working with Moon Il Seok. Together with Detective Im Seung U, she works to uncover the truth behind the murder and its connections to Moon Il Seok and Cho Seo Hui. (Source: MyDramaList)