Soldering wires directly to male AC prongs
Use a supply intended to be wired to

As you can tell, "wall warts" are not intended to have chassis wiring hooked up to their inputs.  An easier and safer approach, provided you have the small amount of additional chassis space (3" by 2" by 1.1" for the Mean Well linked below vs. 2.8" by 1.9" by 1.25" or so for a wall wart) and the ability to bring a ground conductor (3 prong cord) into what you're building, would be to use a chassis mount enclosed supply, such as the Mean Well RM-15-12 (which is under $8 in onesies via Mouser).  These supplies have screw terminal inputs that are intended to be chassis wired to and screw holes for mounting screws so they can be attached solidly to the case instead of flapping around in the breeze like your wall-wart would.