These two petitions are distinct but related. We pray for God's kingdom to come to and among us, and for this to come about or be made known by God's will being done on Earth even as it already is done in Heaven. What is the coming of God's kingdom and the doing of God's will? We'll get to that in a minute.
But first I'm going to digress for a moment to talk about everyone's favorite topic: grammar. The Greek of the New Testament has verbs that change their mood or mode by changing their spelling, rather than by adding extra words. English speakers have to go through more work to use our verbs in all the moods available to us: You go to the store (indicative, stating information); you would go to the store if you could (conditional, stating an if-then scenario); I prefer that you go to the store rather than me (subjunctive, stating an open possibility or desire); go to the store please (imperative, stating a command or direction).
English used to be a little more flexible than it is now, and when our traditional version of the Lord's Prayer was translated, you could do a lot of this just with the order of words. "Thy kingdom come" could mean "that your kingdom might come--we'd like that" (subjunctive) or "Come now, kingdom" (imperative). We don't really use our language this way much anymore outside of church. In prayer, it's easy to fall back on a softer way of saying this. "May your kingdom come; may your will be done." But the original Greek version is in the imperative mood: "Come, Kingdom! Be done, thy will!" or maybe "Make your kingdom come and make your will to be done."
For Luther, the coming of God's kingdom means "whenever our heavenly Father gives us his Holy Spirit, so that through the Holy Spirit's grace we believe God's holy word and live godly lives here in time and hereafter in eternity."
And God's will being done means "whenever God breaks and hinders every evil scheme and will--as are present in the will of the devil, the world, and our flesh--that would not allow us to hallow God's name and would prevent the coming of his kingdom...."
I would prefer to expand on that, so that we see God's kingdom and will being made real to us not just in our own faith and in our own behavior but in the flourishing of human well-being around us, and the sanctification of the world through our actions and prayers.
When I wrote about prayer in my book Sacred Signposts a few years ago, I dwelt on these early Christian prayers and what they may have meant to those praying:
I’m still not sure what effect these people intended when they assembled for their work. When they prayed “Thy kingdom come” or “May grace come and may this world pass away,” were these words meant to hasten the completion of the Messianic age, shouts to nudge the wicked world’s mountain slope toward an avalanche? Or were those words a way of transporting themselves to that end, to the kingdom, to begin to dwell there as a present reality even in the midst of a not-fully-redeemed world, however long the fullness tarried in coming? Does prayer change the world’s timeline, or shift the course of an illness? Or does prayer change us?
Our reason should therefore be moved to seek God, the force of desire should struggle to possess him and that of anger to hold on to him, or rather, to speak more properly, the whole mind should tend to God, stretched out as a sinew by the temper of anger, and burning with longing for the highest reaches of desire. Thus indeed we will be found to be giving God worship in every way in imitation of the angels in heaven, and we shall exhibit on earth the same manner of life as the angels in having as they do the mind totally moved in the direction of nothing less than God