How to build the simplest 12V 2A switching power supply?
First things first - a SMPS is not a simple circuit, and they just get more and more sophisticated. So one option you have is to start with a toy implementation with an inductor, a BJT, a capacitor and a microcontroller to drive the BJT. You wont get 2A out of it on your first go, and you'll probably fry a few BJTs, but you'll learn a lot.

But if you're keen on building something useful, then you need to pick an IC. It's hard to find much simpler than TI's Simple Switcher series. The LMZ14202H would probably meet your needs.

Both the inductor and the switcher are integrated, so you only have to concentrate on the feedback circuit. Follow their application notes very carefully and you should have something running fairly quickly. The LMZ14202H only comes in an SMD package though, so you'll probably want to get a 0.05" breakout board too for prototyping.

Trying to re-use components from existing power supplies is unlikely to be  a fruitful path - those components are likely to be quite specialised, chosen to suit a specific design, and reverse-engineering a commercial power supply design is harder than building one from scratch.