The real problems with Katarina
I will be upfront about this : I dislike Katarina. I dislike playing against her, I dislike playing as her and -- if she wasn't rather absurd in many cases -- I would dislike even having her on my team.
Katarina has seen a rather high increase in popularity of late (at least for my games) and while it has caused me quite some irritation, it has also given me the rare opportunity for insight, given me the chance to ask the ultimate question : "why do I dislike Katarina so much?". It hasn't been the shortest road -- I've analyzed her kit, her numbers, her identity, just about everything I could think of -- but I ended up considerably wiser in regards to her, even if somewhat against my will by having her in so many games.
Taking a look at Katarina's numbers, you'll notice that they're actually fairly low in everything except her ult.
Q : 75 / 115 / 155 / 195 / 235 (+60%) W : 40 / 75 / 110 / 145 / 180 (+25%) E : 60 / 85 / 110 / 135 / 160 (+40%) R : 350 / 550 / 750 (+250%)
These bases are actually considerably low in comparison to other mages and the ratios aren't super amazing. Her ultimate is basically where all of her damage is loaded on paper, and that's incredibly inconsistent because she's easily interrupted during it.
However... the paper doesn't tell the whole story.
Katarina's animations allow her to use her entire kit effectively instantly. In the span it takes for a normal mage to do one of their spells, Katarina can do all three of her basics + the first tick of her ult.
This comes up to 575 (+125%) damage in your face nearly instantly, not even including her ult. All of her basic skills also reset fully with her passive.
Katarina isn't scary because of her ult, she is scary in spite of it : she functions just fine without her ult and it's quite simply the overkill on top or the cleanup button. Even without considering that a good portion of Katarina's damage is going to be completed because of the cast times, reaction times, and often travel times of the CC in question, the CC is only going to stop her ult, not the instant 575 (+125%) single-target damage, which is what really kills people in a majority of cases.
The other problem that not many people may notice is that she has one of the longest range targeted gapclosers in the game : 700 casting range, and there is only one other champion that also does it instantaneously (Talon). 700 casting range means that there is effectively never a time that an ADC -- or even most APCarries -- can deal damage to her without the threat of a near instant 575 (+125%) damage, to say nothing of if she kills another teammate and uses them as a springboard : Katarina extends her zone of threat considerably far without much counterplay beyond "don't be low enough to die" and "don't even try to get close enough to deal damage". All of this while functioning perfectly fine with only one stat (AP) : her passive supplies free CDR, she has no resource to speak of, and her Shunpo assists with her resilience (not that she doesn't build Abyssal / Zhonya's anyway).
I can't imagine how she would function if her animations were overly lengthened. I can't imagine the nightmare it would be to try to lower her damage below what it is even now to be balanced without being worthless. But I can imagine a Katarina with a shorter gapcloser so that she's actually putting herself in noticeable risk to try to do everything instead of extending her zone of threat beyond the enemy's. I'm of a mind that 625 range -- the range on Zed's ult, for ease of visualization -- would be a much better alternative and easier to balance around.
TL;DR -- reducing Shunpo's range would bring her more in line without gutting her terribly.