I figured it out myself.
I did some extensive research and eventually decided HTOUCHINPUT was part of something called the Windows SDK. The AppGameKit help doesn't tell us we need it, but I found it in Program Files(x86)\Microsoft SDKs. I don't remember ever downloading it, so it may come with Windows. Regardless, why do the help files not even mention it if it's necessary?
I put the Include folder into the include directories as well as the Lib directory into the linker and it all works fine now, albeit with a whole heap of warnings.