Ok I have got somewhere. This is a solution that worked for me, having problem from iOS 8 Launch, and being desparate for a solution. You just have to have a Mac to do it. Do it with your own risk.
Open terminal and paste:
ln -s ~/Library/Mobile\ Documents/ ~/Desktop/Real\ iCloud\ Drive
This will create a shortcut of the "real" iCloud drive on your desktop. This means that in this folder not only you see everything that Apple approves you to see but also everything that is hidden from you.
Open the folder named "Real iCloud drive" and select the list view. No other view will work. Consider some time in order to let your Mac sync all the files with the cloud, so let the folder open on the finder for some time.
When everything appears, find the folder named "Settings". Be careful: this folder might be translated to your language (mine was Ρυθμισεις in Greek).
Press the arrow in order to show the folder contents.
Inside there should be a folder called "Dictionaries". This must be the folder causing all the keyboard lag. And yes, you have to delete it. You have to delete it again and again, as long as it needs in order to keep the problem fixed.
In my experience, I deleted it and used my iPhone in order to delete once again the buggy shortcut. As long as I deleted it I tried to write something in the notes. I had still problem! I saw on my Mac that the folder reappeared and I deleted it again, and I deleted again the shortcut from my iPhone. I used the iPhone and it is OK a day and more now!
Also, I had to do the same procedure also for my iPad. I did it an hour or something after I realized and made sure my iPhone worked fine.
Now there is no buggy shortcut in neither of my devices and there is no keyboard lag, with iCloud open of course!
I hope it works for you also!