Why does God permit sickness?

The most direct answer to why God allows sickness is because it is his will to permit it. In theology we have what's called the decretive will of God and the permissive will.  God decrees certain things to happen like the creation of the universe, the flood, etc.  In this he is directly involved and causes them to occur. But, with the permissive will of God we say that he permits things to happen like an automobile accident, or a thief stealing from someone.  With these sorts of things, God is not causing them to happen.  He is allowing them to happen.

Another reason that God permits sickness is because sin is in the world.  Sickness is a result of the fall which was a sinful rebellion against God.  Sin entered the world through Adam (Rom. 5:12) and since we are all in Adam (1 Cor. 15:22) we are all subject to sickness.