She asked me to take care of the baby while she was away.
Literal
She [topic-は] absence-during, I [to-に] baby [possessive-の] care [object-を] do [indirect command-よう] asked.
留守中 (るすちゅう, 'while away') uses the ~中 suffix. するよう頼んだ is an indirect command pattern: ように marks the requested content. 世話をする ('to take care of') is the standard caregiving expression. に marks the person asked.