"looking after the household" means taking care of or being responsible for the occupants and maybe also the property itself. Apart from its obvious literal meaning, "looking into" normally means "investigating". It is not clear in what sense Leela would be "looking into" the household.
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Mr. TomWith both parents gone, Leela, only 14 at the time, was looking after the household. (all the people in the house)That seems rather old-fashioned. This is more recent: