I love this match-up. It's so much fun to play out IMO. While it's very momentum based, both characters can easily break momentum going against them and change it to their favor. I personally believe this either 5-5, or 45-55 from Ike's prospective, but lean heavily towards 5-5.
At first glance, one would believe this match-up is Diddy Kong's favor. He has faster attacking speed, a projectile, faster movement, and the bananas, why wouldn't it look that way? However, this is not the case.
While Diddy Kong has the advantage in speed, he looses quite heavily in range, disjointed hitboxes and power. Ike's Fair effectively shuts down all of Diddy Kong's aerial approaches with room to spare, maybe even Nair stops all of Diddy Kong's aerials but I'm not positive on that one. Fair also beats out Diddy Kong's ground approaches while remaining safe thanks to IASA frames.
Of course, people are probably asking "Why the heck would a smart Diddy Kong approach Ike? He can just stay back and play the peanut spamming, banana controling game and force Ike to approach." And the answer to that is the fact Ike can also take control of at least one banana rather easily, which disrupts Diddy Kong's game plan. Think about what happens if a banana is on the ground, and both Ike and Diddy go for it. Ike's dash attack comes out quick, propels him forwards a step, and hit a very large range in front of himself with a lingering hitbox. It would beat out Diddy Kong's cartwheel or aerial attempt to nab the banana back. Or Nairing to grab the banana, same thing would happen.
One might also point out "What on earth can Ike do with a banana? It locks him to just B moves, which are all VERY situational. Again, this is true, but Ike can indeed use the banana effectively. First of all, he could just throw it downwards onto the ground and play keepaway. Not the most effective thing, but it's an option. Secondly, he could just throw the at Diddy Kong. This can work, but Diddy could also just catch the banana. It's a gamble, but a very dangerous one in Diddy's case. If he gets hit by it, Ike will punish him for it, but hard. Another option is assuming something like Dash Attack was used, and Diddy is off stage, in which case could use it in an attempt to gimp Diddy. Hey, a free item to screw up Diddy's jetpack is a free item.
But Ike's best option is DACIT. I think most Ikes have seen the video where Kirk DACIT'd with a Diddy's banana, hit him, and then nailed him with a reverse Fsmash as he flew by. (A link would be nice incase one of the Diddy Kong players hasn't seen it yet) Ike can do many things out of DACIT: regrab the banana, jab to pivot grab to Bthrow off the stage, Usmash, Fsmash, Ftilt, Utilt, Counter, all of that while sliding extremely far. The only down side is the running start required to get it going, which is thankfully not very far, but enough to give Diddy Kong a small heads up.
Moving on pass the bananas: Diddy Kong can gimp Ike fairly easily, but what else is new. >_> QD is blah blah blah jump in front of it, Aether is drop a banana into it/use an attack with at least some amount of disjointedness. Ike can somewhat gimp Diddy's Rocket Pack easily, as if it needs charging it gives Ike enough time to set up for a Dair or walk-off Fair. Just watch out for the flying run away rockets.
Jab is effective against Diddy Kong as usual, stops his flying peanuts and any bananas thrown your way, ect. Ike will likely KO Diddy Kong with something like Jab -> Utilt, Full Jab combo -> edgeguard, something out of DACIT, Uair due to the fact Diddy Kong can't do much if Ike's below him or Diddy Kong making a mistake. Diddy Kong will likely get his kills with gimps, or one of his smashes, though those kills won't be until later %s.
For stages, ban FD. For neutrals, I usually aim for Pokemon Stadium. The changing of the stage can really disrupt banana positioning, and I like to DACIT during the stage transformation while the smoke is along the ground. Not only does it kinda help hide the fact I have a banana in my hand, the Diddy Kong player may (hopefully) be not 100% focused on Ike, but on how he's going to lay out his bananas for this stage, or how there positioning maybe be changed. That last bit is all speculation on my part obviously. Battlefield as also usable, yays for a superior platform game.
For CPs, Pirate Ship is a no brainer. Avoid something like Brinstar or Norfair. Diddy Kong can control the top platforms while the lava is in the way and screw Ike over more then the lava will screw Diddy Kong's game over. You could be gutsy and try PS2. The wind and electric fields would seriously hamper Diddy Kong's game (IMO, correct me if I'm wrong). Rainbow Cruise is also a good option for Ike, as he should be winning the air battles due to his range.