Hi, Richard
I saw your question title and misunderstood that you were studying Chinese. However some of the Japanese words are original from chinese characters, so I ll just share the chinese meaning of these characters with you, hopefully you ll understand more.
"姫" has three meanings in Chinese:
a complimentary term for women used in ancient China
a title used in ancient China for a concubine
a professional female singer
"王女" can be seen as"王"+"女", so it means either the daughter of the empror or the woman of the empror.