What is the ruling on the excess amount of money the questioner received to pay someone's bills, which that person refused to take back, and is this amount considered a bribe?

It is not permissible for an employee to take money from clients in exchange for their required work, whether it is called a gift or anything else, because the Prophet, peace and blessings be upon him, said: "Gifts to officials are ill-gotten gains." This includes what clients offer to employees for preferential treatment, favoritism, or to expedite their work. The money mentioned in the question is considered a bribe, and it must be returned to its owner. If returning it is not possible, then it must be given in charity.