What should the mother do with the money and gold left with her by her deceased friend, knowing that the deceased did not specify to whom this trust should go, and should she hand it over to the deceased's daughter only, especially since the daughter knows about the existence of this trust and is requesting it in full?

Your mother must hand over the deceased woman's belongings to her heirs, because they have become part of their rightful inheritance. It is not permissible for her to give them solely to the daughter, especially if she predominantly believes that the daughter will take them for herself, as that would be a betrayal of trust. Allah the Almighty says: "Indeed, Allah commands you to render trusts to whom they are due."