Jesus was sent, among other things, to proclaim "the acceptable year of the Lord" (verse 19). What does that mean? It is almost certainly a reference to the Old Testament "year of jubile" in which those who had been forced to sell possessions were to have them returned.
And thou shalt number seven sabbaths of years unto thee, seven times seven years; and the space of seven sabbaths of years shall be unto thee forty and nine years.
Then shalt thou cause the trumpet of the jubile to sound on the tenth day of the seventh month, in the day of atonement shall ye make the trumpet sound throughout all your land.
And ye shall hallow the fiftieth year, and proclaim liberty throughout all the land unto all the inhabitants thereof: it shall be a jubile unto you; and ye shall return every man unto his possession, and ye shall return every man unto his family.
A jubile shall that fiftieth year be unto you: ye shall not sow, neither reap that which groweth of itself in it, nor gather the grapes in it of thy vine undressed.
For it is the jubile; it shall be holy unto you: ye shall eat the increase thereof out of the field.
In the year of this jubile ye shall return every man unto his possession.
The basic idea behind the year of jubile was that if you had to sell your property to someone to pay your bills, you were to get it back in this year. Slaves were freed and could go back to their families. Debts were canceled. Liberty was to be proclaimed to all the inhabitants of the land. Note that this was linked to the "day of atonement."
I do not claim to be able to prove this absolutely, but it seems that Christ was likening his soon-coming atonement with the day of atonement in the year of jubile, at which time people were to get their lost possessions back. In the verse at the start of this discussion, Jesus said that he had come to proclaim deliverance to the captives. Then he said that he was sent to proclaim the acceptable year of the Lord. In the day of atonement in the year of jubile, liberty was to be proclaimed throughout the land. In other words, the captives got to go free that year. I don't think it's too much of a stretch to conclude that Jesus was likening his coming atonement and its resulting liberty to the captives to the year of jubile and its day of atonement.
What's this have to do with healing? Recall that healing was a covenant right even under the Law of Moses. Everyone had a right to be healed by Jehovah Rapha, the Lord who Heals You. Obviously, many people in Jesus' time were not walking in this right. Healing was their rightful possession (Jesus called it the "children's bread"), but they had lost it. Jesus went around restoring this possession to the people who had lost it. His atonement provided the means by which we can have our possessions back. We can get back what Satan has stolen from us, including our health.
Also recall that Jesus declared sickness to be bondage to the devil (see Luke 13:11-16), and Peter said that sickness was the oppression of the devil (Acts 10:38). Given that Jesus was declaring liberty to the captives, he must have been declaring healing to the sick, since he considered it a form of captivity. He came to set at liberty those who were bruised, just as all men were set at liberty in the year of jubile after the day of atonement. I consider this allusion to the year of jubile to be another indication that Christ's atonement was to bring liberty and healing for all.