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Ryouko Mikado is a intergalactic alien doctor who works as school nurse at Lala and Rito school.
As school nurse, Mikado takes care of the usual bumps and bruises that kids usually suffer (although Rito becomes somwhat of a regular in her office). The other half of her career is that of an intergalactic doctor that treats just about anyone that crosses her path. Mikado is not just a medic however, she does display skills in engineering and other such areas by constructing a body for a ghost (manga only). She is also very well connected, having "contacts" that can get whatever her little clinic needs on a moments notice.
As she is an adult and undoubtbly the owner of the largest bust, Mikado is somewhat of an item to the staff and students at the school. In the anime even Rito has a "vision" or "fantasy" or whatever you want to call it, involving Mikado. Although the cast member that is most enamored with her is Saruyama. Also due to the simple fact that she is an adult, the ecchi scenes involving her are few to nonexistant. Although at one point on a beach she instructs Rito to help her apply her sun screen, stating that he needs to get used to skin to skin contact (considering how much skin Rito actually touches I would say that Mikado was right on the money).
Her plot line is mostly in the background for the majority of the manga, with the notable exception where some goons from an intergalactic mafia want her services to create super-soldiers. However in both manga and anime it is Mikado that supplies Rito with the vaccine for Lala's space cold (even though in the anime Rito never has the opportunity to apply the medication).
Mikado acts (with the absence of Rito's actual mother) as a matronly figure, filling another typical ecchi stereotype. That of the mature, very well endowed, mother figure. She seems to have a very deep understanding of how Rito's mind works. As when Lala is suffering from the space cold and acting all shy and modest, Mikado calls Rito on finding Lala more alluring when she is acting shy. Mikado might have been playing matchmaker but it is never mentioned again.