Who should get healed?

This may sound like an unusual question. But God says he makes the rain fall on both the sinners and saints alike.
Matthew 5:45 NIV
He causes his sun to rise on the evil and the good, and sends rain on the righteous and the unrighteous.

Is healing any different?

Modern medicine has helped many, many diseases. In my working career as a doctor, I have seen a small but significant number of illnesses benefit from major breakthroughs which are rightly celebrated. Apart from this, the availability of care and medical help is a great blessing in our country.
Medical care is not determined by faith, though there are disparities in our country, as well as probably all countries. As Christians in the health field, we are in an excellent position to care for those who would otherwise miss out, regardless of faith, level of education, cultural background or any other factor.

The healing ministry in the church is probably a lot wider than what we think. How many countless prayers are silently prayed for loved ones suffering illness? What of the countless Sunday morning prayer lists in all sizes, styles and denominations of church across the globe? 

Are these "ordinary" prayers heard by God and answered?

For some Christians, it is important that God miraculously heals today, apart from and independent to medical science. Some refuse to visit a doctor, even a christian doctor. And for other Christians, there needs to be scientific medical proof before any faith healing is believed.

But I want to mention the “miracle” of our human bodies, with inbuilt healing qualities/abilities that are truly amazing.
As an infant we learn to speak and walk and other normal skills, but we never learn to heal, it just happens. We graze our knee, and are blessed with the gift of pain, the gift of crying, and then the gift of coagulation, the gift of fibrin and scab formation and then the gift of re-epithelialization. Full healing occurs over a few short days without our intervention, skill, or even thought. And there are countless internal mechanisms keeping all of our organ's functioning, which we are totally unaware of. How many cancer cells do our immune system destroy before, eventually, in the occasional person, one takes hold?
For some Christians, there is a need or even a demand to see God at work in miraculous healing.
There are scriptures that state strongly that Jesus’ followers will do greater things than he.

John 14:12-14 (NIV) Very truly I tell you, whoever believes in me will do the works I have been doing, and they will do even greater things than these, because I am going to the Father. 13 And I will do whatever you ask in my name, so that the Father may be glorified in the Son. 14 You may ask me for anything in my name, and I will do it.

How many lepers did Jesus heal?
Maybe 20? No more than 50 from my quick recollection. But modern medicine, now, has a cure for leprosy.
How many people of faith were involved in the discovery of such cures. I actually have no idea, but more than we imagine. I am aware of stories of breakthroughs in science coming to those who pray. And what of the current leprosy mission, https://www.leprosymission.org/ staffed by people of faith, offering funded treatment to those who are unable to access this?
Do these things “count” or are they too ordinary to satisfy those promises?
I’m not sure that I have ever personally seen a dramatic miraculous healing. I believe they happen. I pray for people to receive such healing. But I accept that God may answer these prayers in ways that are unexpected.  I have seen many of my patients recover when they were considered unlikely to. In my understanding, prayer is part of this. God is part of this. He can and does strengthen parts that we already have, such as our immune system, or the wisdom of those making decisions. To me this is as strong a work of the Holy Spirit, as a blind person receiving their sight. But maybe faith is needed to see this!

Hebrews 12:12-13 (NIV) 12 Therefore, strengthen your feeble arms and weak knees. 13 “Make level paths for your feet,” so that the lame may not be disabled, but rather healed.

Naaman had to wash in the Jordan river 7 times.

Sometimes, we have to do ordinary things to receive healing, like submit to the expertise of people who do not share our faith. Is this God letting us down? Or is he giving us opportunity to relate to others we otherwise would not?
In all things God works for the good of those who love him. We believe this, especially when our prayers are met with what some might interpret as silence.
And what of partial healing? I suffered a significant disc protrusion earlier this year. A number of people were concerned for me and prayed for me. Those payers were answered. I am much better, but I still limp. To me, I am as good as I would ever hope to be, and I am thankful. Is my disc still protruding? Probably yes. Is that a blessing? In my eyes, yes.

And what about later in life, should we expect ongoing healing forever? When is it OK to allow nature to take its course? We all must face physical death up to he time of Jesus return. Our Saviour faced suffering and death head on. He could have prayed and legions of angels would have rescued him, but he did not save himself. He accepted death. There was a greater need than his need to stay physically well and alive. It was our sin. He died for our sin. And we know that probably all the apostles died a martyr’s death. And today there are more martyrs than in the first century. How does that fit with an expectation of instant prayer healing?
Stephen the first martyr was a great example of how we are to face suffering.

Am I anti-miracle-healing? 

No.
But I see it rarely, and I do see the disappointment and questioning of sincere believers when prayers for healing are seemingly unanswered. I do not believe it relates to a lack of faith. Faith as small as a mustard seed can move mountains.
God has a time and purpose for everything, including healing and suffering, and disability. Our need to have things fully healed may be due to our fear of illness and disability, or our fear that God may not be with us in these things, or our fear of dealing with people who do have permanent disabilities, or our fear that God is not in fact strong, mighty, present, and able to do these things easily.

Love drives out all fear. 

We should continue to pray for each other, including for our healing. Paul tells us in Thessalonians to pray without ceasing. If we are really in need, I think fasting is worth a go, and I am also into kneeling.

So who does God heal?

Psalm 103 tells us that he heals all of our diseases.

Who is he speaking to?

Everyone.