What is the order of distribution for my deceased mother's estate, and what are the legal shares for each of the heirs (including the children of my sister who died during my mother's lifetime, and other heirs), given that there is a written will for non-heirs?

Your mother's estate is to be divided after discharging all rights, debts, and legally binding bequests (provided they do not exceed one-third). Bequeathing to grandchildren is recommendable but not obligatory. The sister who passed away before her mother does not inherit, as the condition of the heir's survival after the deceased is not met. The two daughters receive two-thirds, and the remainder goes to the brother's son by virtue of 'asaba (agnatic relationship). It is recommended for the heirs to give to needy relatives when dividing the inheritance. Inheritance matters must be referred to Sharia courts to verify all rights and heirs.