(Bonus points if you can identify where the names are from. I'm sorry, but I've never been very good at coming up with names or new characters, so I'm just going to have to borrow some. If anyone has any objections, please send me a message with a better suggestion for the names and characters, and I will take it into consideration.)
Life in the palace was a gilded cage. Constantly surrounded by loyal and obedient servants, you can have anything you want. However, you can't get it yourself. Someone else has to go and get it for you. Even the simplest of tasks, like dressing or eating required the presence of a servant.
Even to Lelouch who'd known this and only this for his entire life, it was ridiculous. Still, there was no helping it. For all his power and authority as a prince, the servants refused to budge on the matter, insisting that performing such tasks for himself was unseemly and beneath him. And enduring such a lifestyle was already beginning to leave its mark upon the prince, the roundness in his face, a layer of softness on his limbs and arms, and most of all, the roundness forming from his middle.
However, there was one thing that gave Lelouch hope for the future, and that was selecting his knight. Every member of the royal family was required to have at least one personal knight. This knight would be entirely devoted to them and to them alone, a constant servant and companion. Finally, someone lasting, a real face among all the ever shifting masses of servants. Someone who could speak freely with him and would actually stay with him so he could develop a real connection.
Normally one doesn't get one's knight until one is at least fourteen, but Lelouch had found a loophole. Since he was only seven he couldn't officially knight someone, he could however designate someone as his future knight. This would place them essentially on the fast track, as they would be provided with special privileges and training that they would need to mold them into the perfect warrior and knight. Further although it wouldn't have the same authority as an official summoning of an official knight, no one would deny him if he asked to see his potential knight in a free moment. And although they still wouldn't be a full knight and thus not a constant companion, they would be a companion no the less.
Thus the reason why the chubby young prince would be visiting the Knight's Academy, to observe and select the one he would choose as his knight.