When the believer in your example prays, he (or she) is acknowledging his weakness and inability before God, and calling out to God for help. In so doing he is drawn closer into relationship with God. Praying in this circumstance provides practice of thinking about God in the right way, which can a have a lasting effect on the state of mind the one who prays. This can occur before any result of the prayer becomes apparent.
Beyond this, God can and does answer prayer, and when he does it reveals his good nature and strengthens the faith of the one who prayed. To refuse to pray in this circumstance is to shut God out, and to reject relationship with him. As in your example, not praying does not mean that the person will not get better, but it is harmful to the one who does not pray.
Many Christians will be able to give you examples of answered prayers, and many base their faith, in part, in this experience of the action of God.
Of course prayer does not work like commanding an genie in a bottle, but that is not neccessary for prayer to fulfil its purpose.