She was kind enough to show me all around the city.

Literal

She [topic-は] kindly [emphasis-も] city-inside [object-を] here-and-there guided-for-me.

あちこち ('here and there, all over the place') is a demonstrative compound from あちら ('over there') + こちら ('over here'), compressed into a casual adverb. It adds a sense of thoroughness — she didn't just show one spot, she toured the speaker around multiple places. 市内 (しない, 'within the city') specifies the scope of the tour.