Many fully-grown adults require a certain amount of noise to sleep (hence the sleep function on your TV and on many alarm clocks that include a radio function as well). I did find one study (and admittedly only read the abstract) that showed that background music can result in delayed sleep and sleep with less depth. However, the study was more focused on whether music helped the listeners to stay awake after sleep deprivation. It also had a very small study group, so the findings may not be all that reliable anyway.
Many sleep training books actually suggest using white noise for babies and toddlers (which I realize, your daughter is not, but I know of adults that also use such devices) that are learning to self-soothe to help prevent waking, but white noise is quite different from the talking, sound effects and musical combinations one might hear while the TV is on. Perhaps adding a fan for its noise, a musical device that can play recordings of waves, or rain or whatever into her room, or a white noise device could help a little as a way of transitioning from needing to hear the TV to just to needing a little background noise that is soothing to her.
Counter-intuitive, but. . . I gather from the statement:
Often, when I go to bed at a reasonable hour, she wakes up and comes
into my bed
That it isn't really about the noise as much as it is about knowing where you are or what you are doing. If that is the case, maybe going in and giving one light kiss on your way to your own bed would be a nice way of communicating that information to her. "I'm going to bed now, but I'll just be in my bedroom after this point." If she isn't awake, she might still pick up on the cue sub-consciously and feel more secure. If she is awake, she is getting the info she needs to feel more comfortable and fall asleep anyway? Might be worth asking her what she thinks of the idea at least.
You've asked another question that makes me think you are already getting the kids to work on staying out of your room (or that this question is retro-active) so I'll leave that part of the equation to that other question.