I would ask the sitter directly. You mention only that your child knew all about these shows, and while I do agree with you that it seems very unlikely that she would know every theme song and the shows' running order unless she had watched them while with the sitter -- is there any chance at all that she has seen these shows at another kid's house? Is there another child whose house she visits fairly frequently? Kids are sponges and soak up things like theme songs fast.
This is why I would just ask the sitter point-blank: "When X was put on the TV, Suzy knew the theme song and the order of shows, and I know we have never watched any of this with her. Please tell me, have you had the TV on at all while you're sitting her?" Then see what she says about it. She might deny it flat-out, and it's going to be hard to prove it if you feel she's lying. Or she might say, "Oh, it was only on the background for the music, I wasn't really letting her watch it" etc. Then you have a choice to make about how you handle whatever response she gives. Disabling the TV/cable/whatever during her hours in your house is certainly an option, I guess, but I'm not sure why you should go through that hassle if you were very clear with her about no TV, and she acknowledged that -- did she?
If you feel she's really a good sitter, I would give her another chance and say that you want to be clear that the TV is not on at all during her sitting hours, period, and that it is important to you as her employer that she respect that condition even if she disagrees with it. I would emphasize the "as her employer" part because even if she "feels like one of the family," she is still your paid sitter--the TV isn't. I don't know that I would fire her over one infraction on the no TV rule, or even two, but you and your husband should think this through: Is there a point at which you'd be willing to fire her if she did it? If it's that important to you, you might have to make that stand.
Is your one child the only one she's sitting at these times? Are there other kids around? Is it possible she's sitting too many kids at once, or uses TV as her break time or lunch time? If she says yes, the TV is on at times, explore with her why she's got it on--she might need the break, or might be overextended.
If she sits your child in her own home, did you and she discuss a no-TV policy wherever your daughter is, including her own home? If she said, yes, I get it and won't let your daughter watch TV while we're together, and that applied to her home -- I'd hold her to it. You're not dictating to her how to behave in her home; you're holding her to the terms of her employment.
I get the aversion to TV for young kids. Our friends' son was in a day care that the boy just loved and mom and dad were reluctant to pull him out but finally did so because they wondered why he came home talking up Batman this and bang-bang that, and they dropped in one day unannounced to find the kids sitting with glazed eyes in front of cartoons aimed at much older kids. This wasn't even a home-based day care but a facility. They pulled him out because they were not paying someone to plop their son in front of a television set, whether that set was showing Batman or Dora.