What is the ruling of Islamic law on leaving a sick mother who prefers to stay in her home, and moving to another residence 2 kilometers away from her?

The rights of a mother are immense, and it is obligatory for children to serve and care for her, especially in her weakness and old age. If you are able to provide a servant for your mother in your traveling sister's home, while committing to visit her and check on her well-being, then there is no harm in you moving. However, if you are unable to do so and fear harm to her by leaving her alone, then do not leave her.