She would need to agree because the conditions of khul’a are that the wife has aversion and she gives a badhl. Moreover, a new marriage contract would need to be enacted. Thus, if you would like to remarry her then you have to determine her willingness.

A new marriage contract would need to be enacted for you and her. For what you heard, it is for a different situation, someone who has divorced his wife and returned/remarried her twice, then upon the third divorce there is a requirement of her marrying another man (and intercourse), getting divorced from him, and then she can marry the first husband again.