The phrasing of this verse “the time cometh speedily that Satan shall have no more power...” indicates that these are events of the last days. The millenium is that period of time when Satan shall be bound, and these events are attached to the period immediately preceding that day:
“And he laid hold on the dragon, that old serpent, which is the Devil, and Satan, and bound him a thousand years.” (Revelation 20:2)
“For Satan shall be bound, and when he is loosed again he shall only reign for a little season, and then cometh the end of the earth.” (D&C 43:31)
“The Millennium consists in this--every heart in the Church and Kingdom of God being united in one; the Kingdom increasing to the overcoming of everything opposed to the economy of heaven, and Satan being bound, and having a seal set upon him. All things else will be as they are now, we shall eat, drink, and wear clothing.” (_Discourses of Brigham Young_. Deseret Book 1977, p.115).