。
She looked after my dog for a month.
Literal
She [topic-は] one-month-duration my dog [of-の] care [object-を] saw.
面倒を見る ('to take care of, look after') is the canonical idiom for caring for a person, pet, or sometimes a project — literally 'to see/watch over the trouble.' Don't decompose: 面倒 by itself can mean 'bothersome, troublesome,' but in this idiom the unit reads as 'caretaking.' 一ヶ月間 with the suffix 間 marks the duration ('throughout the month'), more emphatic than plain 一ヶ月. 私の犬 with の possessive frames the speaker as the dog's owner.