Is it permissible for my mother to undergo dialysis in mixed-gender centers, and is it permissible for her to go there alone for that purpose?

If there is no dialysis center specifically for women, there is no sin upon a woman who needs it to go to mixed-gender centers. If female nurses are not available, there is no sin upon a male nurse to perform the procedure. There is no sin upon her to go alone if she is safe from temptation and unlawful situations, such as the presence of trustworthy female nurses who prevent khalwa (definition) (private seclusion with a non-mahram (definition) man). The Lawgiver has removed hardship from His followers and linked obligation with ability. In cases of dire necessity, looking at and touching for treatment are permissible, even the private parts, so that prohibition does not lead to hardship. It is permissible for a man to be in seclusion with two trustworthy women who observe proper hijab (definition), and this is the relied-upon opinion.