Is the woman obligated to care for her elderly mother, who requires special care and daily injections, causes her great psychological stress, and affects her ability to care for her children, given her sisters' refusal to participate in the care?

You are not obliged to bring your mother to live with you in your house, especially if that would cause you harm. All her children must care for her, and you must treat her kindly to the best of your ability and apologize to her for what you are unable to do.