This verse has long struck me. I know of a certainty the answer to the first two questions. Is there a balm in Gilead? yes. Is there a physician there? yes. Why then is not the health of the daughter of my people recovered? That question is a little more difficult.
Though I think it has something to do with what happens on our end of Christ's Atonement. He will not heal us or save us against our will. We are only able to partake of the blessings of the Atonement as we try to follow Christ. When we repent, we can recover. When we act on our faith and serve, we can recover. The reason that the health of many is not recovered is because they will not apply the balm nor will they even ask the doctor for it.
The gospel of Jesus Christ is the only thing that will truly and fully heal us of all things.