。
She kept talking despite the noise.
Literal
She [topic-は] noise [without-minding-にかまわず] kept-talking.
~にかまわず ('without minding, regardless of, paying no attention to') is the negative te-form of 構う ('mind, care about') in classical-flavored ~ず form. ~にかまわず is a formal/written set phrase, often used to describe persistence in the face of distractions or obstacles. 話し続ける ('keep talking') is a compound verb 話す + 続ける ('continue'); the productive ~続ける auxiliary attaches to ます-stems for 'keep doing X.'