Is it permissible for a sister who was leading the prayer (imam) to complete her prayer after her ablution (wudu) was broken and then renewed, and is the prayer of the mother who waited for her considered valid and accepted, and do they both have to repeat the prayer?

If the Imam's prayer becomes invalid, the prayer of the congregants (Ma'mum) does not become invalid, according to the majority of scholars. Therefore, your mother should have completed her prayer when your sister invalidated hers. If the sister performed Wudu' before her mother finished praying and wished to join her, that would be permissible.

Regarding your fatwa that the sister should perform Wudu' and then return to complete the prayer, with the mother following her for the remainder, this is valid according to those who validate the prayer of a congregant who says the Takbir (Allahu Akbar) before the Imam. Some have used as evidence the Hadith of the Prophet ﷺ when he remembered that he was in a state of Janabah (major ritual impurity) before saying the Takbir for prayer. He ordered the Companions to wait, then he performed Ghusl (ritual bath) and returned and led them in prayer. This Hadith is authentically reported in the Sahihayn (Bukhari and Muslim) and other collections. Its wording in Muslim indicates that his departure was before the Takbir. As for the narrations that suggest his departure was after the Takbir, they are not in the Sahihayn and are not free from criticism. Some scholars interpreted this as meaning he stood in his prayer place and prepared for the Takbir.

Based on this, if your sister acted upon your fatwa and returned and completed the prayer with your mother, their prayer is valid, and they do not need to repeat it according to this view. However, it is preferable to avoid this in the future to avoid scholarly disagreement.