Why doesn't the keyboard appear to type name or number in Contacts? I can't add a new contact as keyboard doesn't appear in the Contacts. I have deleted email account from Settings. Also the Contacts service is on in iCloud.

I have an iPhone 4 on iOS 7.
Delete any other keyboards you're using like emoji etc then it should be fine.

I was facing the same issue and it resolved only when I deleted the keyboard.

Go to settings, general, keyboard and then look if there are any extra keyboards there.