When I click the "Play" button the launcher vanishes like normal, but then quickly pops back up and Minecraft never starts.

Checking the development console I see this error:
Exception in thread "main" java.lang.UnsatisfiedLinkError: C:\Users\Born\AppData\Roaming\.minecraft\versions\1.7.9\1.7.9-natives-6358202749934\lwjgl.dll: Can't load IA 32-bit .dll on a AMD 64-bit platform

I have searched for a couple days now on how to resolve this. I've uninstalled all Java, and reinstalled it all from scratch. Done the same for Minecraft, and in every order.