Is it permissible to travel without a mahram or the father's permission to reach a sick mother after ten years of deprivation, based on the principle of "necessities permit the forbidden"?

A woman is not permitted to travel without a mahram. Try to convince your father to allow you to visit your sick mother, and tell him about the long period of separation from her. Seek the help of close relatives who can influence him. If he agrees, then find a mahram to travel with you, even if it requires paying him a fee. Increase your supplications and adhere steadfastly to the commands of Allah.