I too have experienced similar issues - slow keyboard response, high CPU (> 90%) consumption, very poor battery life, Message/Settings would take 7 - 10 secs to open, some apps take too long to open, some crash etc., etc,..... 😠 😟 😮
As suggested by several forum users, I tried several combinations of turning OFF several SHOW PIECE features ... finally narrowed it down to CALENDAR Notification ... turn it off in ...
Settings > Notification > In Notification Center (scroll down to) > Calendar > Notification Center [OFF].
Turning this off defeats the purpose ... 2 steps forward (with iOS 5) and 4 steps backwards (Calendar alerts/notification is very important). With Calendar Notification ON, the CPU usage is over 90% ... the moment it is turned OFF, it returns to less than 10%. After this finding, I turn ON Notification during my work time and OFF while not ... not something I like, but have no other choice 😮
Well, this is my experience ... it may or may not fix for others though!
I use the app System Status to monitor the vital statistics of iPhone/iPad. Once you install System Status, you might also want to have a look at the the System Log. Open System Status > Details > System Log. Its got a wealth of information for technically inclined users. I found a line that said something like ... "Caldav notification trying to open <something> .... Tried 20 times unsuccessfully ... error ... etc,. The same thing will repeat every minute or less ... so I decided to turn Calendar Notification off.
NOTE 1: My iCloud is ON for Mail, Reminder, Bookmark, Data/Docs & Find-my-iPhone and so is sending Diagnostic Data to Apple (let them still get debug data in the hope that they will send the iOS 5.0.1 sooner). My CPU usage is between 7% - 15% and battery life is acceptable.
NOTE 2: I don't have this problem in my iPad 2 running iOS 5 🙂 no issues with Calendar Notification .... and I sync both iPhone 4 & iPad 2 with the same iTunes Windows 7 (x64) PC and I use iCloud between these 2 devices.
Cheers