Is it permissible to perform the obligatory Hajj on behalf of my mother, who has previously performed 'Umrah but is unable to travel because she is caring for my sick brother?

It is not permissible to perform Hajj on behalf of a living person unless they are incapacitated, meaning they are unable to perform Hajj themselves due to an enduring disability, such as old age or an incurable illness. Therefore, your mother is not permitted to delegate the obligatory Hajj due to your brother's illness; rather, she must perform Hajj herself when she is able.