Understanding where the constant $2/N$ comes from in Fourier transformation
The two comes from the fact that a real valued sinusoid is really the average of two complex one.

$$ \cos( \theta ) = \frac{ e^{i \theta} + e^{-i \theta} }{2}  $$

There is the "2" in the denominator.  In the DFT, the other half is located at bin $N-k$, if $k$ is the bin number.

That definition of cosine comes straight from Euler's equation:

$$ e^{i \theta} = \cos( \theta ) + i \sin( \theta ) $$

The $N$ comes straight from the definition of the DFT.  The relationship beteen the amplitude, N, and the magnitude of the DFT bin is why I like to use the $1/N$ normalized form of the the DFT over the more conventional unnormalized form.

I recommend that you read my first blog article The Exponential Nature of the Complex Unit Circle for an understanding of Euler's equation and my blog article DFT Bin Value Formulas for Pure Real Tones for understanding the bin values of a pure tone and what "leakage" really means.  Equation (19) is the answer to your "big question".

The $\Omega$ comes from the continuous FT.  The DFT is the Discrete Fourier Transform.  Too often, they are conflated leaving confusion like yours.  It takes a lot of heavy math to understand the relationship between the two.  The premise of my blog articles is that the DFT can be learned and understood straight from the summation definition without any reference to the continuous case.

Hope this helps.

Ced