What should one do who kept the motor of an old woman's washing machine that broke down more than ten years ago, and he cannot reach her or her relatives?

It appears you fell short in returning the engine to its owner, and this is a clear error. You must first repent to Allah, then strive to find out the woman's whereabouts to return her right to her. If you are unable to do so, then you are obliged to donate the engine or its value on her behalf. If you come to know her later, then return to her the equivalent of her right, unless she permits your charity.