I translated "自分自身の力" as inner strength. "分自の力" by itself is sufficient to imply "own power", as in "to do something under your own power", so "自分自身の力" feels as if it has a stronger emphasis on the individual, which I thought might be captured by "inner strength" rather than just "own power" or "own strength". This has the feeling of "digging deep into her own resources", in this case her "inner" resources.

"She has to rely on her own power" sounds a little unnatural in English. People tend to talk about "inner strength" rather than "own power", unless you are talking about "someone doing something under their own power". In this case the translation would be "She had to do it under her own power". However, if you translate it this way, then you lose some of the meaning of "頼". You would only be able to judge from the context which was more important, e.g. she was not in a position to rely upon someone else, or she needed to find the abilities within her self to solve a problem. If it is the former, then the current translation may be a good fit, if it is the latter then "She had to do it under her own power" might be better.