Don't just spam AS on Diana. You won't live long enough against good players to use all of it.
As for Lich Bane, I will often go with that instead of Nashor's as my last major item. Gives you more up front burst which = doing your job better.
Since they toned Diana down, she is not the faceroll instagib whoever you want and never die demon she used to be. So while her passive is a huge damage source, you either don't have the survivability combined with sufficient AS to play around it. The best way to play is to burst someone down, hourglass to survive and get your CD's back up. Then either keep fighting or back off and dive back in. You just can't afford to sit in the thick of things as much anymore until after to have mogs/sunfire/randuin's. --- LoL IGN: alpha10