thoughts on Gödel's incompleteness theorems

I've been trying to understand Gödel's incompleteness theorems for over seven years. A few months ago I really dug in and understood the mechanics of the proof. But what does it mean? This is a frustrating thing about modern mathematics. Hermann Weyl complained about this very thing, that verifying every step in a proof does not necessarily lead to insight. I get how Gödel's proof works; I can lead you through the coding of sentences into numbers, the simple though long process of defining the analog within arithmetic of deduction within the formal system, the construction of the sentence which "refers to itself," and the ensuing double-contradiction from assuming that sentence is provable or its negation is provable. Yet I remain unsatisfied. What does it mean about the nature of numbers, and formal logic?
I have certain intuitions about it. There's something about the meanings of negation, universal quantification, and the combination of the two, that needs to be explicated. There's something about how logic tries to go "beneath" arithmetic and fails in that attempt; I think that you can't reduce one to the other, and that both are co-present before the embodied subject (my Merleau-Pontyian, hermeneutical best) who has the freedom to creatively interpret one in terms of the other, and neither has a true primacy over the other. There's something about infinity, and giving it a finite symbol, and acting like it's somehow contained in there while at the same time reasoning with it purely as a symbol (which is obviously self-contained and finite).
I keep thinking about it and I should probably stop for now. Push too hard and the gears grind. Rest, let your unconscious knit the brain into a new shape
It's similar to my frustrated resignation to the proof that √2 is irrational. I know that if it were rational you could reduce the fraction it to least terms and either the numerator or denominator would be odd, but then you show how for it to be the square root of 2 both must be even, and that makes a contradiction. But why? Numbers overflow the bounds of certain reductive methods we have to think about them. Maybe it's truly mysterious. But I want a better answer than that. There's some assumptions within logic, perhaps the static nature of classical logic, and that the language of logic needs to be separate from its domain of discourse for excluded middle to hold, and then Merleau-Ponty's idea of the body as a sensing-sensible which finds itself as a thing among things as the condition of possibility for sensibility, and perhaps the undecidable sentence of Gödel is such a sensing-sensible which disrupts the assumptions of classical logic. But I can't work it all out without what mathematicians derogatorily call "hand waving." Some day I'll come up with something.
Playing one brooding Austrian off another, allow me to quote Rilke at length:

 Leave to your opinions their own quiet undisturbed development, which, like all progress, must come from deep within and cannot be pressed or hurried by anything.  Everything is gestation and then bringing forth. To let each impression and each germ of a feeling come to completion wholly in itself, in the dark, in the inexpressible, the unconscious, beyond the reach of one's own intelligence, and await with deep humility and patience the birth-hour of a new clarity: that alone is living the artist's life: in understanding as in creating. There is no measuring with time, no year matters, and ten years are nothing. Being an artist means, not reckoning and counting, but ripening like the tree which does not force its sap and stands confident in the storms of spring without fear that after them may come no summer. It does come. But it comes only to the patient, who are there as though eternity lay before them, so unconcernedly still and wide. I learn it daily, learn it with pain to which I am grateful: patience is everything!