I wouldn't know against acutal players, but the point of lich bane is to help your burst, because it's and extra 75%base AD+50%AP ratio to your mark proc.
The ideal combo is Q>wait for CD>R>AA>Q>E>AA
Remember, you need AAs to proc mark now.
In bot games, I generally blow my enemies up easily after finishing gunblade and Lichbane. Lich Bane in particular (hell, even just the sheen) really helps your burst.
Remember that LB is 80 AP and some movespeed on top of the spellblade.
So for jungle, just finishing Magus+LB+Gunblade is now 240AP
240*0.5 = 120
And then she gets ~44+2.5lvl from her base AD Let's say you're level 11 (meaning 86 base AD) when you finish these three items?
It's 184.5 extra burst damage, per spellblade proc
More or less an extra E to your burst per proc.
And it keeps scaling up.
Granted you could get a Deathcap for just a bit more gold, it gives you 40 more AP by itself, and then 280*1.3 =84, so it's 124 more AP, 5% less movespeed, 300 more gold, a much worse build path,.
Now a full cycle of your combo, deathcap would give 75.6 more damage per Q(with proc), 25.2 more damage per E and 42 more damage per R.
It also increases your passive magic damage on hit by 14%of your AD, which at lvl 11 with gunblade should be 17.64
So assuming Q>R>AA>E is your full and only combo, that's 75.6+25.2+42+17.64=160.44 damage with deathcap instead.
So yeah, you're likely to get ever so slightly better results with lich bane than with deathcap.
At least before you get even more AP.
When that happens deatchap should slowly outscale lichbane.
and of course, there's always the option of getting both.
Anyway, I don't know about Katarina, but I do know that for Akali, the general order ought to be: Jungle Item(without enchant)->Hextech Revolver->[Finish gunblade+get Sheen+finish lichbane+get boots in whatever order makes sense that game]->Magus->Deathcap->[Void staff+finish boots]
Swap for whatever defensive item you need if you really need it. If you're doing fine without it the jungle item can go. But i'm usually going with trailblazer, so if you're going for something else and still feel like building glass cannon, you probably want to keep Stalker or Skirmisher.
So yeah tl;dr: Lich bane gives 80 AP and does 50%AP+75%base AD per proc.
If you just think of it as a 50% extra AP ratio on your mark proc, that should sound pretty damn good in and of itself really.
Also note I don't pvp much if at all. I'm too timid.
Armchair analysis on the other hand... oooh yeaaah.
P.S. stats based on her base AD and ratios from the wiki, which, y'know, is a wiki.
The item stats i pulled from here so I assume they're guaranteed to be accurate (at least it matched what I remember from ingame).