This fix worked for my phone:
1. Delete the two outlook.com and one Hotmail account from my Mail, Contacts, Calendars (under settings)
2. did a full backup of my phone to my PC (for the first time ever :-) I didn't need to, but I did this just in case.
3. Go to Settings, General, Reset, Reset All Settings. It sounds scary, but other than your wallpaper, passcode and wifi password, all your other content remains intact. I checked my app data after restoring and it was fine
4. After restarting, added a new account, of type Outlook, and entered my settings. And voila, all was good.
NOTE: Mind you, for point 4 above, the apple support representative did say I should add an email account of type "other", ignored this but was lucky it was ok. So, if this doesn't work for you, do another reset and try and add an account of type "other".