"He kindly asked me for help" means that he was kind enough to ask me to help him rather than asking someone else to help him. "He asked me kindly for help" means that he asked me in a nice way to help him. Does "Il m'a gentiment demandé de l'aide" have the former or the latter meaning in French?
I think this can also be translated as "He gently asked me to help him". ..... plus, isn't this more accurate (or literal, at least)?............ The official English translation ("He gently asked me for help") would more likely be translated as "... demande(accent aigue) d'aider" perhaps?