Since there was no reply, he wrote her another letter.
Literal
Reply [subject-が] didn't-exist [because-ので], he [topic-は] once-more she [to-に] letter [object-を] wrote.
もう一度 means 'once more' — もう ('already/more') plus 一度 ('one time'). A straightforward cause-and-effect sentence using ので to link the lack of reply to the action of writing again.