It is the Dragon Quest 2 log in screen (hence the DQ2 password).
The first thing she types in is a resurrection spell from the Dragon Quest series
The Spell of Restoration (復活の呪文, ふっかつのじゅもん, fukkatsu no jumon) is the name of the password system used in the Famicom and MSX versions of Dragon Quest and Dragon Quest II. Similar to other games of the time, the player's accomplishments were converted into a password, which was then written down and entered later to resume the game from the point at which the player left off. With the advent of battery backup (first seen in the series in Dragon Quest III), password systems became obsolete, and the Spell of Restoration was never used again. It differs from the passwords of other games in that it is recognized by some characters as a convention of the game world and not simply a means of resuming gameplay.
"Kami nomi zo shiru sekai" follows the same pattern used, which is 3-3-4:
Dragon Quest II
Due its broader scope (three-member party, multiple flagged plot events, and a wider array of equipment), Dragon Quest II employed a larger 52-character Spell of Restoration: Four lines of 3-3-4, followed by a final line of 3-3-4-2.
As to why she mistypes, it is most likely a reference to the fact you load a life. A life only you know of, hence The world god only knows, Keima's world.