Her hard service which has been completed is the discipline God subjected her to which brought her to repentance and faith. Most versions other than the NIV translate this phrase as "her warfare has ended" This may allude to all the warfare she had experience in the process of being disciplined, like the coming Babylonian captivity. The Septuagint has "her humiliation is accomplished." God humiliated Israel in order to develop in her that most essential of character qualities in a relationship with God, namely humility. This was accomplished.
But before continuing on we must consider the time frame in which Isaiah was speaking. Isaiah was not speaking of his own time. For Judah was yet to go into captivity. So also her sin was yet to be paid for almost 700 years in the future through the atoning work of Christ, which Isaiah speaks of in chapter 53. But God knows the future and for Him it is as if all things have already been accomplished before they take place, as for example in Revelation 13:8 it speaks of Jesus, "the Lamb that was slain from the creation of the world.""Her sin has been paid for"
Who paid for her sins? Many view this to mean that she paid for her own sins through her chastisement. Indeed can a person suffer enough to pay for their own sins? And if so then why would Christ have to die? Under such a scenario there would be no eternal damnation. For even sending people to hell their suffering would eventually pay off the penalty for their sins. Rather Isaiah is alluding to God paying for her sins of which Isaiah will elaborate upon in chapters to follow.
Furthermore this phrase tells us first of all that sin can be paid for and secondly that sin has to be paid for. God does not just outright forgive sin without satisfying the demands of justice, namely that sin be paid for. God who pays for the sins of his people. This concept is unique to Christianity, which is the true Judaism. All other religions which have the concept of sin do not portray God as paying for people's sins. Rather their gods either ignore the issue of payment and grant outright forgiveness, or their gods have their people pay for their own sins.
"double for all her sins" This phrase should end the debate as to whether Christ's atonement was just sufficient for the elect, or whether he paid more than what was sufficient. Calvinists with the concept of "limited atonement" hold that Christ only died for those who would inevitably be saved. But along with that concept is the idea that even with respect to the elect Christ only made a sufficient sacrifice. The reasoning behind such concepts is the idea that it would be unjust to pay a penalty greater than was warranted. But Christ's death was not simply about satisfying justice. It was about revealing the abundance of God's grace.
"The gift is not like the trespass. For if the many died by the trespass of the one man, how much more did Godís grace and the gift that came by the grace of the one man, Jesus Christ, overflow to the many!" Rom 5:15
For God "desires all men to be saved and to come to the knowledge of the truth. For there is one God and one Mediator between God and men, the Man Christ Jesus, who gave Himself a ransom for all." 1Tim 2:4-6