Am I obligated to serve my sick mother by moving in with her, knowing that it would be difficult to reconcile this with the demands of my home and work, or is it sufficient for me to maintain ties with her and serve her as best as I can without sin?

The rights of parents are great in Islam, and serving them is obligatory, especially during times of weakness and illness, even if they are disbelievers. A married woman is not obligated to leave her home to serve her mother; rather, a maid can be hired if the brother and his wife are insufficient to serve her. One should strive to serve the mother to the best of their ability, and the husband should appreciate this.