Presumably by "send the screen to sleep", you mean turn off the backlight. There is little economy to be achieved by shutting down the actual LCD display driving circuit.
I would think at least five seconds would be more reasonable. In any case, it is easy enough to do given that you are either using a backpack which provides control of the backlight, or a shield which does (bearing in mind the design fault described in one of the "sticky" threads here) or you are controlling it with a transistor of your own arrangement.
Your "loop" code if written correctly, would allow for a countdown timer which is continuously advanced to 3000 (or whatever value you decide on) beyond the current "millis()" value on any keypress, and when in the loop, that value is exceeded, the backlight is turned off.
If you need more description than that, it is your turn to post your code for review.