Who kills Appia?

Appia is killed by Ababio, Densu's guardian. Ababio is a local power-broker, a very important man in the area, who is also very dangerous. He's not the kind of man you'd ever want to make an enemy of. Yet that's precisely what Densu does by choosing to pursue the path of healing instead of engaging in manipulation and exploitation like Ababio.
Ababio recognizes Densu's great talents and abilities and wants to use them in the service of the colonial authorities. Appia is a threat to this plan. The young crown prince has come under the influence of his mother, who in turn has been influenced by the healers. As such, Appia has turned against the ruling whites. If he is permitted to succeed to the throne, Ababio will lose all his power and influence, and he simply won't allow that to happen. That's why he has Appia murdered. At the same time, he frames Densu for the crime as punishment for turning down his offer to become the next king. In getting rid of Appia and setting Densu up as the patsy for his murder, Ababio is trying to kill two birds with one stone.