While the Studios used to have working soundstages where things were filmed (the 90s edition of the Mickey Mouse Club - you know, the one that launched the careers of Britney, Christina, and Justin - for better or worse), those same soundstage have now been retrofitted into the One Man's Dream attraction and Who Wants to Be a Millionaire. They do still have one soundstage availiable for taping, but that will soon contain sets from the new Lion, Witch, and the Wardrobe movie.
So, the only show that I know of that possibly tapes at MGM is Millionaire (the version with Meredith Vierra). Once or twice a year, the show travels to WDW to film 3-4 weeks worth of shows from there. Other than that, no tv shows are taped there (except for the occasional spots during ESPN: The Weekend and Super Soaps Weekend).
Sorry - there's so much else to do in the World that I'm sure your daughter will be plenty occupied!