Try it and find out! An adaptor like that is dirt cheap.
I did have a PS2 mouse that wouldn't work when plugged through a PS2/USB adaptor. I'd heard somewhere that the mouse has to support USB protocols even if its built-in plug is PS2 (I guess the adaptor only changes the shape of the plug, the mouse would still need USB logic inside). If that's true, and also holds for keyboards, you may be out of luck.