Are the jewels left by the old woman with my mother, with her saying, "If I die, they are yours," considered lawful for my mother or not, knowing that the old woman has two daughters?

The jewelry is considered a bequest from the woman to your mother. A bequest is permissible for a non-heir only within the limit of one-third (of the estate). Anything exceeding one-third is dependent on the heirs' approval, based on the Prophet's (peace be upon him) saying to Sa'd: "A third, and a third is much." If the jewelry falls within one-third of the estate, the bequest is executed. If it is more, only one-third is executed, and the remainder is returned to the heirs. The bequest must be proven by evidence or witnesses.