In doing some research on your question I came across this statement by Elder Robert D. Hales, "And God said, 'Wherefore, because that Satan rebelled against me, and sought to destroy the agency of man, which I, the Lord God, had given him, … I caused that he should be cast down.'[ Moses 4:3] Those who followed Satan lost the opportunity to receive a mortal body, live on earth, and progress. Because of the way they used their agency, they lost their agency."
If you can answer the question of how those who followed Satan lost their agency you will understand how Lucifer would have taken away our agency. So, what does it mean to say that the devil and his angels no longer have agency? What is agency, according to the scriptures? Elder Hales said "Recently I received a letter from a friend of over 50 years who is not a member of our church. I had sent him some gospel-related reading, to which he responded: 'Initially it was hard for me to follow the meaning of typical Mormon jargon, such as agency. Possibly a short vocabulary page would be helpful.'
"I was surprised he did not understand what we mean by the word agency. I went to an online dictionary. Of the 10 definitions and usages of the word agency, none expressed the idea of making choices to act. We teach that agency is the ability and privilege God gives us to choose and “to act for [ourselves] and not to be acted upon.” [ 2 Nephi 2:26] Agency is to act with accountability and responsibility for our actions. Our agency is essential to the plan of salvation. With it we are “free to choose liberty and eternal life, through the great Mediator of all men, or to choose captivity and death, according to the captivity and power of the devil.” [ 2 Nephi 2:27]
"Our agency—our ability to choose and act for ourselves—was an essential element of this plan. Without agency we would be unable to make right choices and progress. Yet with agency we could make wrong choices, commit sin, and lose the opportunity to be with Heavenly Father again. For this reason a Savior would be provided to suffer for our sins and redeem us if we would repent. By His infinite Atonement, He brought about “the plan of mercy, to appease the demands of justice.” [ Alma 42:15]"
Satan did not need to remove a physical thing from the spirits who followed him. He did not need to change the makeup of their brains. He did not need to take over the government. He just needed to get them to choose to lose the ability and privilege to act for themselves and not be acted on. To choose to not be accountable and take responsibility for their actions. Once we voluntarily give up our agency it is lost to us.
Satan and his followers will be eternally punished for the choices they originally made, though they no longer have the ability to choose anything else.
Robert D. Hales, “Agency: Essential to the Plan of Life,” Ensign, Nov 2010, 24–27