A short answer: なおします is たどうし. なおります is じどうし.　 In a bit more detail: naoshimasu (なおします) is the -masu form of the transitive verb (たどうし) naosu (なおす). naorimasu (なおります) is the -masu form of the intransitive verb (じどうし) naoru (なおる). In Kanji: naosu = 直す or 治す naoru = 直る or 治る 直る or 直すrefers to inanimate things: to fix, repair, correct. 治る or 治す refers to humans and other animals: to treat, cure, heal.

I didn’t understand about your question because of my English. After seeing Kanji-san’s answer, I understood and I agree with him.