ECDSA: How to retrieve a non-random k
I believe you are being confused by what $k^{-1}$ means.  It doesn't mean "interpret $k$ as a real number, plug it into a calculator, and hit the $1/x$ button".

When we use the $k^{-1}$ notation, what we mean to say is that $a = k^{-1}$ is equivalent to saying that $a \times k = 1$.

You will say "doesn't that mean doing exactly what a calculator does?".  Well, it would, if the multiplication was performed in the field of the reals.  However, we're doing everything modulo $n$ (within the field $GF(n)$, as $n$ is prime), and so the multiplication is done modulo $n$; hence we really have $a \times k = 1 \pmod{n}$.

In this context, when we write $k^{-1}$, we really do intend the modular inverse.  And, when we derive $k = \frac{z_1 - z_2}{s_1 - s_2}$, we're not ignoring the modulo operation; it's been assumed implicitly; what we really mean is $k = (z1 - z2) \times (s_1 - s_2)^{-1} \bmod n$, there $(s_1 - s_2)^{-1}$ is the multiplicative inverse (modulo $n$) of $s_1 - s_2$.

When you derive $k = \frac{z_1 - z_2}{s_1 - s_2 + n*(i-j)}$, well, since everything is done modulo $n$, including the internal additions, we effectively have $n=0$ (because, modulo $n$, those are the value), and so $i$ and $j$ drop out, and we're left with the original derivation.

Since you explicitly asked questions, I'll answer them (even if they ought to be obvious once the above has been clarified):


  Mathematically, is there another way to retrieve $k$, even if the $k^{−1}$ part is really large? Maybe using some inverse modulo?


Yes, an inverse modulo is precisely what you're looking for.


  Why is it that many implementations use the inverse modulo instead of $k^{−1}$? (I've seen it in two libraries so far)


The inverse modulo that the libraries do is what is meant by $k^{-1}$


  Why doesn't it work when I change the implementation to use $k^{−1}$?


The mathematics of ECDSA assume that the math works out in one specific field (the field $GF(n)$); if you do the computations in a different field (say, the field of real numbers), well, it won't work.