The question I have is, how does the berachah stand on its own when we don't add any prayers? We ask HaShem to grant our prayers, but we don't pray for anything.
I think the key is in this phrase: (ומלפנך מלכנו ריקם אל תשיבנו) "from before you O King, don't return us empty handed." It's possible to read it differently, "and within us your kingship, don't return us empty." This plays off of Mosheh Rabbeinu's request to HaShem that He fulfill His promise to us and dwell within the nation. (How does "from before you" mean "within us"? The root of the word is panim, which means within, and it is conjugated to mean within you, so how does it mean within us? Because when your Kingship is revealed within us, it becomes a place sanctified to you. It becomes your place.)
That is the heart of the berachah: HaShem should answer our deepest prayer which is don't leave us empty of your Holy Presence. (If we look long and hard enough at our tefillot this is always the backbone, the underlying foundation, that HaShem should dwell within us.)
The concept of HaShem dwelling somewhere is a complicated one, as we know HaShem permeates the whole of creation. (ממלא כל העולם כולו) But, as we explained in another post, the divine presence of HaShem represents a revellation of HaShem's light which is otherwise normally hidden.
When we pray, we are praying that HaShem's influence on the world should be revealed within us. It should be clear to us, and we should make it clear to those around us. HaShem is King.