Kei to Ayakashi

For generations, the Shiranoi family has made a livelihood as exorcists drawing on the power of the Ayakashi that resides in a sword. However, in childhood, the protagonist, Kei, was unable to unsheathe the sword, and her father, Riichi, spurned her as unworthy of becoming the next head of the family. But hidden within her father's actions is a certain secret...

(Source: Shogakukan, translated)