You reach this boss by finishing Kartos's mine cart challenge in the casino world. I.N.E.P.T. takes offense to Kartos's distribution of a Pagie, and attacks.
This boss, I.N.E.P.T. is a little bit tougher than the other previous World bosses. It takes place on a circular mine cart track, and has three phases, which can get a little tricky. You have to keep whaling on him with cannonballs, but you can only fire straight ahead, and the curved track prevents the cannonball from reaching him.
Phase 1
I.N.E.P.T. will plant bombs on the rail. Simple enough, at first - just jump over them. Once you've done so, I.N.E.P.T. will attack head-on. Fire a cannonball using
and IMMEDIATELY JUMP. Merely hitting I.N.E.P.T. will not stop him from following through on his attack, but you can jump to avoid taking damage.
After doing this a couple of times, I.N.E.P.T. will start being clever with the bomb placement. When you see two bombs slightly closer together than normal, DO NOT TRY TO JUMP THEM TOGETHER. Blow the first up with a cannonball, then jump the second. Two or three more iterations, and we begin Phase 2.
Phase 2
I.N.E.P.T. will now abandon bombs and just repeatedly try to ram you. He'll alternate ramming from in front and from behind. Straightforward enough - just jump him. Unlike before, however, whenever he's in front of you, fire cannonballs as often as you can. The curvature of the track no longer prevents them from hitting.
After a couple of rounds, he'll again get clever. Note the number of times his eyes flash red before jumping. If it's three - jump. If it's two, he'll move in close and pause for half a second before attacking. Jump at that pause. Do that and hit him some more with cannonballs, and we're on to Phase 3.
Phase 3
I.N.E.P.T. will start firing missiles from behind you, preventing you from attacking with your cannonballs. The missiles always fly over your head and land ahead of you on the track, then turn around and attack you. Treat this the same way as the bombs: jump. The missiles will follow the rail and blow up I.N.E.P.T. When he starts grouping them together, jump and accelerate
to jump over the lot of them. A few rounds, and I.N.E.P.T. will concede defeat.