Does my sister's watching of Korean "idols"—which mostly means "beloved" rather than "worshipped"—with her intention being to enjoy them, not to worship them, count as disbelief or worshipping other than God?

You have done well in advising your sister and clarifying the ruling on watching TV series and listening to music. We advise you to continue counseling her with wisdom and good admonition, and to pray for her guidance. The word "idol" means an adored one, an icon, or a beloved one. Your sister watching that series does not make her an unbeliever or a worshiper of other than Allah. However, this does not mean it is permissible to watch it.