It took Toph one visit to the institute before she agreed to take over the job of keeping an eye the former Fire Nation princess. Zuko made the solemn request after threats to, not his life, but Azula's.
There was no one he trusted more to protect those important to him, and while Toph had been nervous at first (because who really wants an insane war criminal on their hands?), the hollow vibrations that rolled off the former prodigy was not the only thing that swayed the metalbender's mind; it was the memory of who she had been.