Even trying to persuade my sister to give up was useless.
Literal
Younger-sister [target-に] give-up [so that-ように] persuade [even if-ても] useless was.
~ように説得する ('persuade someone to do X') uses ように to mark the desired outcome. ~てもだめだった ('even doing X was useless') expresses futility — the persuasion attempt failed regardless of effort.