What is the ruling concerning a sum of money I was entrusted to deliver to a partner after the dissolution of a partnership, but he refused to take possession of it, absolving the first party, and the first party refused to take it back, knowing that the money decreases annually due to zakat? And is it permissible for the partner's wife to take it without his knowledge to solve his financial problems?

If the money is rightfully due to the man who left work and refuses to take it, then you should either keep it as a trust until he accepts it or until he dies, at which point it transfers to his heirs. Alternatively, you can hand it over to him in the presence of witnesses to clear your liability, and it is preferable to hand it over to the Sharia (definition) court.

However, if the money is a voluntary offering and not rightfully due to him, then he is not obligated to accept it, and you can keep it as a trust or hand it over to him.

Handing the money over to his wife does not clear one's liability except with the owner's permission. Zakat (definition) is obligatory upon the owner of the money and cannot be paid on his behalf except with authorization, because it requires intention. Therefore, you are liable for the Zakat you paid out without the owner's permission.