You spent half your document doing Ohm's and Kirchhoff's Laws which didn't do anything to explain op-amps.

Then, rather than trying to explain the gain function, you dive back into the same confusing analysis. You don't define k and shortly after - you declare "I hereby drop ≈ and just use =."

ARRRRRRRRGH!!!!!!!!!!!!!!!!!!!!!!

There are many, many much better articles online for learning about opamps. Seriously, if you want to understand an opamp, look elsewhere.

And - you didn't even explain gain for anything but DC. The word "gain" was used only once.

What is k? It is the open loop gain!

What do the resistors you add to the opamp do? They bound the open loop gain to a function. Using the figure on pg8, R1 is the input resistor and R2 is the feedback resistor and the gain of that circuit is R2/R1.