The pronouns may be confusing you. Both pronouns in "He shall see His seed" are Jehova. "His seed" are all who are born again of the water and the spirit (John 3:5) who are in Christ; they become God's spiritual children. Look at it in the YLT version.
"10 And Jehovah hath delighted to bruise him, He hath made him sick, If his soul doth make an offering for guilt, He seeth seed -- he prolongeth days, And the pleasure of Jehovah in his hand doth prosper.
11 Of the labour of his soul he seeth -- he is satisfied, Through his knowledge give righteousness Doth the righteous one, My servant, to many, And their iniquities he doth bear."
Christ bore all of our sins in His death on the cross, thereby allowing all who will believe, repent, and be baptized into Christ to have remission of sins, and we are then counted for the seed of Abraham.
"26 for ye are all sons of God through the faith in Christ Jesus,
27 for as many as to Christ were baptized did put on Christ;
28 there is not here Jew or Greek, there is not here servant nor freeman, there is not here male and female, for all ye are one in Christ Jesus;
29 and if ye [are] of Christ then of Abraham ye are seed, and according to promise -- heirs." (YLT)
When we are in Christ, God sees Christ's sacrifice covering us. And as counted for the seed of Abraham we become the Israel that is saved with an everlasting salvation.
"But Israel shall be saved in the Lord with an everlasting salvation: ye shall not be ashamed nor confounded world without end." (KJV)
Those who are in Christ are "His seed", the children of God.
(Bold emphasis is mine.)