I'll state up front that I'm not real fond of fan fiction. But then, I'll confess that most of the examples I've encountered have dealt with characters and settings with which I'm not familiar. For a true fan, I can understand why revisiting familiar territory from a new perspective might be satisfying, though for the writer, appropriating a cast of characters and a ready-made universe created by someone else always felt to me like a kind of cheat.

One problem stems from the need to refer back to the original source material in order to establish context for the current story. That happens here, where we're told several times and in several different ways that Korra just recently saved Republic City from Kuvira. This, of course, meant little to me beyond the stock superhero meme, but I have to wonder if a fan would need the referential material at all, since one assumes they already know it. In any event, such referential material comes under the more general heading of back story, which is never to be confused with the story proper.

Rule #1 (and 2 and 3 as well): Stories happen now. Which brings us to another potential problem with fan fiction. I actually read one story that billed itself as Sherlock Holmes fan fiction, a genre that has been visited often by many writers both serious and otherwise. This particular story dealt with the period prior to the actual Holmes narratives, detailing Dr. Watson's prior military experiences. It was a good story, actually. Good enough that had the main character been named Dr. Smedley, or Dr. Grossinger, the story would have suffered not one bit. In other words, the "fan" connection was an affectation, grafted on to no apparent purpose.

I will say that you have created a story of sorts for Korra and Asami, and while I have scant knowledge of the events leading up to the present timeline, a good story will draw in the reader anyway, offering the present as a sufficient focal point, letting the past filter in between the lines. You've done that, to a point. Unfortunately, your story in the present isn't much of a story. But it could be.

Asami wants to take Korra out for a celebratory dinner to commemorate her recent victories, but for Korra, this will constitute a "coming out," so to speak, allowing the general population to see the two of them as a romantic couple. This is a theme that certainly has the potential to extend far beyond the parameters of a cartoon series from Nickelodeon, but you don't really deal with it. In fact, you seem almost embarrassed to mention it at all, filling the opening paragraphs with conversation about topics that both of them know, and each off them knows the other knows. In fact, the "problem" is shown to be no problem at all, as Korra enjoys such celebrity and good will, the population is more than willing to accept her under on her own terms. In other words, it's a phony problem, one that is immediately dispensed with, after which they enjoy pleasant meal.

To be completely fair, I acknowledge that I might have it all wrong. Were I to watch some episodes of the original show, I might find that you've captured precisely the tone and perspective in a way that will immediately resonate with your fan readers. I'll leave it to you to decide if that is or is not the case. All that aside, however, you should keep in mind that a story is not about a particularly event, or a specific problem. Stories appear when characters confront the difficulties created by such situations, and are forced to break out of whatever status quo state defines their condition in order to resolve the problem. This usually entails decisions and actions, which often result in unintended consequences and unforeseen implications, necessitating new decisions and actions, etc., etc., etc. The problem here is that nothing was demanded of Korra. She's the main character, but she needed to simply wait patiently to have the true situation revealed to her. One might argue that she actually did plenty (the aforementioned salvation of Republic City) without which the good will of the populace would not have been as much in evidence. But that all happened in some other story. You can't channel back story for energy in the present. That can only come from the dynamics in the present. You have the potential here; you just need to exploit it.