To find this Secret Mission, you have to re-enter the room where you first encountered the Beelzebub (in addition, you must also have collected the Blue Orb Fragment from the preceding hallway). You can do this during Mission 7 or in the interval between Mission 7 or Mission 8, however; if you do the latter, it will not count towards your Devil Hunter Rank at the end of the mission. If done during Mission 7, your health will constantly decrease because of the Guiding Light you're holding.
To win this, you have to get one Kyklops to kill the other without dealing any damage by your hand. If you deal any damage yourself, regardless of the final hit, you will fail. Despite what the secret mission description says, you only need one Kyklops to kill the other: After that, you can polish off the second on your own (or even leave the room). You can stand between them, jump when they attack, and hope that one eventually kills the other with their rock-spit attack. Another, safer way to do this is to jump onto the pipes jutting out of one side of the room, and let them blunder around until one dies. Then jump off and collect your reward (in some cases this method will not work).