I'm guessing you already checked in-game to make sure controller mode wasn't enabled.

Are you running FFXIV through steam? If that's the case, it's possible that the steam overlay is bugging out. Barring that, I would try uninstalling/reinstalling/updating your keyboard drivers from the device manager. If you have a gaming keyboard that uses third party drivers, that may well be causing the issue too.

I recall a player was having issues with mouse/keyboard input lag while using dual monitors, but he resolved by closing the twitch stream he was viewing.

If none of that works, you might also try removing the client's User Configuration files from the system; they might be bugged.