Translation:She asks not to watch the TV while she is sleeping.
The subject is она. There is no "me" mentioned in the sentence. If you mean who is the sentence being addressed to, it could be anyone. If it was addressed to "me" it would read "Она просит меня не смотреть телевизор, пока она спит." Смотреть is the infinitive so it doesn't tell us anything about who is speaking or being addressed.
Probably a better way to translate this would be "she asks that the tv not be watched while she sleeps."