Have you also looked in the battery compartments of the mouse & keyboard?
Not an original HP one as far as I know.
You can order the Logitech Unifying Receiver and connect it.
If you already have one, you could pair it with the keyboard.
Logitech Unifying Receiver
Yes, with my last wireless keyboard I could wake up the PC by pressing a key. And yes, with these multimedia keyboards, the FN key usually has to be pressed together with the function key. Without FN, the keys are used for media navigation (play, pause, etc).