Kiss the Villain to Survive

The system gives her its terms on her first day: kiss the fake eunuch or die. She has just woken up inside a novel as a minor character, a girl from a ruined family who is in love with the crown prince, and the prince has already broken off the engagement and slapped her. So she kisses the eunuch in front of everyone. He holds power over the whole court, and people there say he is mad. He is also a fake eunuch, cold to everyone in the open and warmer than that underneath, and after the kiss he begins to fall for her. She keeps herself alive by earning favor points from him. Along the way she gets his family out of the danger they are in.