How do Mormons interpret Daniel 2:44?

The LDS Church's "Old Testament Student Manual" has a good, if small, compilation of statements on this subject with some commentary. Search for the question "How Is It That the Kingdom Set Up by God Will Consume the Other Kingdoms?" on this site.

A short answer to your question before diving in: Mormon teachings, as far as I have been able to find, do not indicate a specific event leading to the destruction of nations. That the kingdom of God will have real political power after Christ's coming is found in Mormon teachings, though the exact process by which this political power is gained is not specified to my knowledge, though it is associated with the great growth of the kingdom of God as it rolls to fill the whole earth.

There are two relevant quotes found in the manual. From Elder Bruce R. McConkie:

During the millennium the kingdom of God will continue on earth, but in that day it will be both an ecclesiastical and a political kingdom. That is, the Church (which is the kingdom) will have the rule and government of the world given to it. (Bruce R. McConkie, Mormon Doctrine, p. 416.)

So, the "rule and government of the world" will be given to the kingdom of God. Exactly how is not indicated, but we know that when Christ comes again He will reign as the king of kings and lord of lords over the whole earth in both a political and ecclesiastical sense (e.g., Revelation 20 which states that Christ will reign for a thousand years, or the LDS Church's Tenth Article of Faith "Christ will reign personally upon the earth").

President Brigham Young taught:

The Lord God Almighty has set up a kingdom that will sway the sceptre of power and authority over all the kingdoms of the world, and will never be destroyed, it is the kingdom that Daniel saw and wrote of. It may be considered treason to say that the kingdom which that Prophet foretold is actually set up; that we cannot help, but we know it is so, and call upon the nations to believe our testimony. The kingdom will continue to increase, to grow, to spread and prosper more and more. Every time its enemies undertake to overthrow it, it will become more extensive and powerful; instead of its decreasing, it will continue to increase, it will spread the more, become more wonderful and conspicuous to the nations, until it fills the whole earth.” (In Journal of Discourses, 1:202–3.)

This quote describes the kingdom of God as already possessing political power, and as it grows and becomes "more wonderful and conspicuous to the nations" it will fill the whole earth, at which point one can say that its intrinsic political power has taken over all nations.

As far as whether specifically the United States of America will fall, President Ezra Taft Benson often taught of the necessity of preserving the United States of America as a place of freedom, and in particular of preserving the United States Constitution. The following quote, from a talk entitled "This National Shall Endure", indicates that President Benson is sure that the United States will "endure", though he does not indicate a specific timeframe for this endurance.

May we do all in our power to strengthen and safeguard this base and increase our freedom. This nation will, I feel sure, endure. It is God-ordained for a glorious purpose. We must never forget that the gospel message we bear to the world is to go forth to the world from this nation. And that gospel message can prosper only in an atmosphere of freedom. We must maintain and strengthen our freedom in this blessed land. (Ezra Taft Benson, Brigham Young University Devotional Address, 4 December 1973)

That the LDS Church believe the Constitution is an inspired document can be found throughout President Benson's teachings, as well as the 101st section of the Doctrine and Covenants. Church leaders (such as President Benson in the talk linked above) often acknowledge the importance of the USA and its freedoms in the establishment of the LDS church, and in this way one can think of the Constitution and the similar documents of other nations (and, more broadly, the nations themselves) as belonging to the kingdom of God in that they are associated with establishing it, even if indirectly.

These quotes unfortunately do not have the specificity to match the specificity of your questions. However, I do have some thoughts on this subject that have been informed by my lifetime of growing up in the LDS church that, although I don't have specific references to point you to. I've always been under the impression that after the Second Coming of Christ, those who are left will recognize Christ as their lord, savior, and king. Some will recognize this immediately, others who are less familiar with Christ's teachings will become convinced of this over time. At this point, the political authority that existed before Christ will become irrelevant, because everybody will recognize Christ as their ecclesiastical and political head. In this way the kingdom of God will have broken apart all other kingdoms and filled the whole earth, though through the peaceful conversion of individuals rather than through any sort of hostile takeover or destruction. This seems very consistent with Brigham Young's statement quoted above.

I also remember watching a video where President Benson stated that when Christ comes again, he was confident that flag of the USA would be flying in the breeze. However, I have not been able to find that video or text since, so do not take that as fact yet. I post it here in case someone else is aware of a reference for these words and can share it with me.