Read it again: Isaiah 53:5 KJV But he was wounded for our transgressions, he was bruised for our iniquities: the chastisement of our peace was upon him; AND WITH HIS STRIPES WE ARE HEALED.
Jesus giving up the ghost didn't pay for sin. Jesus paid for the elect's sins by drinking the cup of the wrath of God, which started in the garden of Gethsemane.
Matthew 26:42 KJV
He went away again the second time and prayed, saying, O my Father, if this cup may not pass away from me, except I drink it, thy will be done.
The unsaved will have to drink that same cup of God's wrath, which means they are alive when they drink it, as Jesus was alive.
Revelation 14:10 KJV The same shall drink of the wine of the wrath of God, which is poured out without mixture into the cup of his indignation; and he shall be tormented with fire and brimstone in the presence of the holy angels, and in the presence of the Lamb: