I'd like to use the rest of the lesson time to practice the example sentences in the textbook until I have them completely memorized for the Eiken.
ご質問ありがとうございます。
"I'd like to use the rest of the lesson time"=「レッスンの余りの時間を使いたいです」
"to practice the example sentences in the textbook"=「教科書の例文を練習するために」
"until I have them completely memorized for the Eiken."=「英検のために完全に暗記するまで。」
ご参考までに。