There are actually two sentences here and from a grammatical point of view, they are both correct. However, there is inconsistency in two areas. First, the verb tenses are different. For consistency, the verbs should be in the same tense. Second, the wording "does his request" does not sound as smooth as it should because parallelism is broken. In light of this, the sentences can be written in the following ways:
"The little child asked his mother to wash his face. She quickly complied."
"The little child asked his mother to wash his face. She quickly agreed."
Or if you wanted to keep the original wording, you could say:
"The little child asked his mother to wash his face. She quickly fulfilled his request."
Both are grammatically correct, but the first options have better coherence better.