How can one deal with the children of a deceased sister, for whom she left a will, and who demand absolute trust and freedom of action while engaging in worrying behavior, all while preserving the mother's will and looking after their best interests?

The mother's will regarding guardianship is not valid because it is the right of the father. As for custody, it belongs to the closest in rank as long as the conditions are met.

If the children reach an age where they are feared to be corrupted, then the father or whoever is capable of protecting them from corruption has a greater right to their custody, and this is especially true for daughters.

The claim of trustworthiness made by the sister's children is contradicted by the daughter's mixing with young men.

What is obligatory upon you regarding your sister's children is to look into their matter carefully; if you are capable of reforming them, then take them and do not pay attention to what they say. If you are not capable of that, then give them to their father if he is qualified for it.