Did she killed her father and that's why she uses his sword or... she is "waiting" him to get her revenge?
The latter. The Spirit and Plague are the two killers who haven't killed anyone.
Her backstory is left ambiguous. But since murders in the other killer backstories are usually explicitly stated it can be assumed that she hasn't killed him yet.
The Devs confirmed in the lore stream that she never managed to kill her father as she got pulled into the realm before doing anything. The fury of not having been revenged keeps her going along...
I think it mentions in the Lore that the Spirit made an oath with the Entity, promising to kill for it in exchange for the ability to murder her father (which is likely not going to happen)
The entity made a deal that he wouldn't hold up on. If the Entity let her get revenge her eternal rage would die out. After that she wouldn't kill. And no amount of torture would get her to kill again. She would be a resting spirit. With that said, Spirits rage is very very powerful. If she were to ever realize that the Entity isn't holding up his side of the Oath she would likely rip a rift in the entity and haunt her father.