When you said you did not have the folder, I checked my laptop, which has never run Sticky Notes. The StickyNotes folder was not on it either. But after starting the application, it was added along with the referenced file.
You might try running it as Administrator to see if it makes a difference.
I do not know for sure what installs Sticky Notes, but my best guess would be with the Tablet PC utilities that are part of the Windows components and can be uninstalled and reinstalled. I know the Snipping Tool and on screen keyboard are part of this group, but not sure about the sticky notes.
Have you opened and administrative command window and typed the following to check your system files?
sfc /scannow