In her book, Kalakaua: Renaissance King, which I have recently finished reading, Helene G. Allen provides a description of the meeting the two, as given by a member of the King's entourage (though she does not tell us which member):
A door opened, and his Holiness, Leo XIII, a thin and spare old man with an extremely pale face, entered and slowly moved across the room, while all bowed in reverence, to a chair on a dais raised a few inches from the floor. In front of him another chair was placed for the King; around the Holy Father the Cardinals were grouped, and we of the suite stood near the King.
The Pope began the conversation at once in Italian, which was interpreted by Cardinal Howard. He asked many questions about the Hawaiian kingdom. The Cardinals joined, and soon showed that they were well informed about the condition of the native Catholics in Hawaii, of whom there were almost as many as there were Protestants. The Holy Father said to the King: "Will you present your companions?" The King presented us. The Pope asked: "Are they natives of your country?" The King replied that we were, and the sons of Protestant missionaries. Cardinal Howard laughed, and said, "Then they are in the opposition." The Holy Father smiled. There was no solemnity in the interview; it was only a pleasant chat.
"Do my people in your kingdom behave well?" asked the Pope.
"Yes," said the King, "they are good subjects."
"If they do not behave," said the Pope, "I must look after them. Why do you have a white Minister in your government?" he continued.
The King could not make a brief explanation and turned to me. I answered, for him, that the kings of Hawaii chose educated white men, who were better able to deal with the foreigners, who held most of the wealth of the country.
Cardinal Howard asked, "Are there any Catholics in your government?"
I answered: "No, the American Protestants entered the country before the Catholics did, and have kept control of public affairs; but no efficient Catholic is excluded from high office by reason of his faith."
There was a pleasant twinkle in the Holy Father's eyes, and he smile while he spoke ...
After an interview which lasted twenty minutes we kissed the Holy Father's hand and rose. He said to the King, "Your country is far away. I shall pray for your safe return" (124-125).