When her neighbor fell ill, she asked the doctor to treat him.
Literal
Neighbor [subject-が] became-ill when [in-に], she [topic-は] doctor [at-に] treat [favor-てくれる] [so that-ように] requested.
~てくれるように頼む is a set 'ask someone to do X (as a favor)' construction. ように marks the content of the request. 治療する ('to treat medically') is the formal Sino-Japanese verb. A typical narrative sentence.