Honestly, you're pretty screwed as an ADC against a fed Akali. Your best bet is after you get your ie/pd/whatever your first two damage items are is to grab some heavy MR. Banshee's Veil is probably best, then buy a Mercurial Scimtar ontop of that. That should allow you to still buy some damage as well as defense.

Or you could get those two items, build a Maw of Malmortius, Mercurial, and Wit's End. There's a lot of AD solutions to buying Magic Resist, I'd abuse it.

Akali suffers from severe energy loss from long engages, making her much easier to deal with by having some survivability. Maw of Malmordius or Warmog's are good. I personally would suggest a Banshee's Veil, as pure health would make her spellvamp heal herself too much, and pure MR won't be enough to save you. It'll also help you spam your spells, which is what Akali's weak to, anyway.

Maw, Wit's End, and Mercurial all are not very good items.

Just buy the damn Giant's Belt and upgrade it to Warmog's. As long as you can survive her initial burst, you'll be able to back away and win. Remember to kite the rest of her team, and hopefully your team can either dive and kill the enemy team or peel off Akali for you.