One is the passage in Isaiah 53:5: "By His scourging we are healed." Ultimately, of course, all the saints are healed, when we are given our heavenly bodies.
Let me very quickly say that God can, and sometimes does, heal. In our church we practice James 5:14-18, and the elders do pray for those who are seriously ill and ask for prayer. Some have been dramatically healed, others have died. If God is sovereign, then He is the one who chooses whether to heal or not. Little wonder that the preacher you referred to denies His sovereignty – that puts man in control, and makes God his servant.
Incidentally, a good friend of mine just went through an experience. He was elderly and gravely ill. To the bitter end they insisted that God was going to heal him. The wife couldn’t accept that he was dead. Their system of belief was that if they had enough faith, God was obliged to heal. How sad that in his dying moments there could only be doubts and fears ("If God will heal me if I have enough faith, then if I’m not healed I must not have enough faith. If I don’t have enough faith to be healed, do I have enough faith to be saved?")
There are several texts that directly address suffering in a way that might be helpful. I have done sermons on all of the texts I'll mention.
First, Psalm 73
Next, 2 Corinthians 1
Also the entire book of 1 Peter
The whole point of the Book of Job is that God may use the suffering of His saints for purposes we don’t even understand in this life. The answer to Job’s questions is the sovereignty of God (chapters 38ff.).