"She herself wanted them to help each other."
Translation:Ŝi mem volis, ke ili helpu sin reciproke.
This is the jussive -u, expressing "a world as it should be", that you often find after words such as deziri or voli. Some languages would use the subjunctive here.
You can't have helpi because there is a subject ("ili"); and you can't use the English accusative-with-infinitive construction where the subject of the verb turns into the object of "want" and the verb stays in the infinitive since that's an English speciality.
And even in English it only works with a few verb (want is the only one I can think of now). Compare situations such as "know" -- you would say "I know that they help one another" and not (usually) "I know them to help one another", and with "see", "I see them to help one another" is not possible at all and it has to be "I see that they help one another". In Esperanto, you have to use this "that" construction also with "want".
So the question is not helpu vs. helpi but helpu vs. helpas, and here the answer is because it's not something that is actually happening but something that is wished for.