She kept talking even after I asked her to be quiet.

Literal

She [topic-は] be-quiet [indirect command-ようにと] I [subject-が] asked after [even-でも] talking continued.

黙るようにと私が頼んだ後でも — ようにと marks the indirect command content. 後でも ('even after') uses でも for concession. しゃべり続けた uses the compound 続ける for continuation.