Okay, I'm going to say this briefly.
#The reason that Ahri is seen as broken by some and balanced by others is because of the concept of windows of power.
Let me explain.
In mid lane (and perhaps in other lanes, haven't played those enough with a critical eye) we dance around each other throwing skills at each other, taking advantages of weakpoints. These are windows of power.
For example, Brand's windows of power are when he has all his spells up. When he uses W to poke, the window lowers, because now he has lost kill potential without his ult and can only do an emergency stun with E>Q. If he uses E to farm the creeps, this is the moment many mid laners attempt to attack him if they haven't gotten hit by his passive. The reason is because Brand's windows of power revolve around having all of his spells up to fight back and mitigate retaliation
For an example more relative to Ahri, let's take Viktor.
The bulk of Viktor's damage in lane comes from his Lazer. He loses trades if he doesn't hit the lazer, therefore Viktor's windows of power fluctuate around his Lazer cooldown
Demonstrated as a chart, these champ's windows of power can be shown as below:
http://www.psychstat.missouristate.edu/introbook/sbgraph/normal0.gif
This is a completely blank and irrelevant graph in terms of its content, but what it represents is a window of power. The high point is when the champ's kill potential is high, and the low points are when they miss their essential skill or use it to farm.
Ahri however has a window of power that can better be described using this graph.
http://forums.serif.com/attachment.php?attachmentid=37436&d=1353529642
If we can see the bottom of the curve as Ahri's lowest point, when all her spells are gone, we can see that Ahri's windows of power are much lower than other mages but are more constant overall.
What I mean is, that if Ahri and Viktor were to duke it out, Either one of them could delete the other. However if Viktor misses one skill, Ahri can kill him, if Ahri misses one skill, she has the mobility required to avoid death or kill him
More plainly, Viktor has a higher potential burst, but Ahri has better reliable burst, especially when she lands Charm.
What this means in laning is that while you can easily predict another mage's moment of weakness and all in them, There is never a point when Ahri is weak in laning (once all skills are up). She will always be able to retaliate against attack, however she trades this for total damage.
Safety is another big thing. Viktor and most other mages have limited mobility, meaning that mispositioning can be lethal, however Ahri has the mobility rivaling assassins.
So why is this fair?
Well, to put it plainly, Unless Ahri is ahead, other midlaners have superior AOE damage and burst Brand, Ryze, and Vik can all kill in teamfights much better than Ahri, however Ahri has the tools required to make single picks, and she's better at this than any one mage.
It's like she took all the AOE damage of other mages and compressed it into a package designed to kill one target than high tail it out.
This brings me to another point.
Ahri is terrible against tanky champs who can eat a full rotation, because her skills are tailored towards taking out a single target, she flounders in teamfights against the front line. She wants to unload her skills into the enemy carry, but if they are protected enough she can't do it. If someone were to attack her, she is forced to use her ult to escape but in the process becomes useless in the teamfight in comparison to her counterpart.
Conversly, Viktor, Brand, Lux, these mages have massive AOE damage and can decimate the back line through the frontline Best examples of whom I listed here are Brand and Lux. Viktor's more of a mid-ranged mage.
So, TL;DR. Ahri trades high AOE damage that is good in teamfights to single target damage focused on making picks and the mobility to do so. People find her frustrating in laning because she doesn't have high or low points as obvious as most other mages, her windows are much closer together but as a consequence never get as high as other mages. As a result people have to engage at her weak moments which are comparatively stronger than other mages' weak moments.
Anyway I hope someone made sense of this pile of rambling.
