There are “guitar keys” which have sharps and “piano keys” which have flats. When you get up into the high numbers of sharps and flats, you get a kind of crossover. Six sharps is F# Major, and six flats is G-flat Major, which are the same key. So this song could be in either six sharps or six flats.
With modern electronics, you can write and play the song in a simpler key like F Major (one flat) or G Major (one sharp) and have it come out in a different key. That may be what happened here. That octave leap at the end just sells it.
Cyndi Lauper: “Girls Just Want to Have Fun