She's the Perfect Vampire?

A princess of the Joseon court fell ill with a condition that had no cure. The king had a forbidden rite performed on her and she became a vampire. A seal set on her at the same time took away the thirst for blood and the weaknesses her kind carries. The dynasty fell. Other vampires learned what had been done to her and came after her body, and she left and kept moving. She did not kill and did not drink from the living. By the present day she was living in a private hospital and taking her blood from bags. Her identity got out. She took a false one and joined Baekya, a unit that hunts vampires. The team leader, Kang Chan-woo, had lost his parents to vampires and worked on the basis that all of them were to be put down. She stayed in the unit under the false name and grew attached to him while the hunts went on.