App not deleting from storage/phone not backing up

hi

my wife's iPhone 5 is playing up. It won't back up (either on wifi to the cloud or on iTunes with my laptop). Pictures are backing up, but not other data from apps or contacts etc... I have read somewhere it might be a rogue app, so I have found the following:

I have noticed I have an app in the 'manage storage' part of settings that won't delete, it is greyed out. It shows version 1 and 174mb of data but no documents. It lets you delete it, but it reappears even if you do a reset of the phone... I can't tell what app it is etc.

So, I have tried loads of things, logging out and into iTunes on the phone, upgrading my iTunes on my computer, resetting the iPhone (not complete restore as I cannot back it up), you name it, I have tried it...

I have been on with Apple numerous times, who have told me that I need to restore the phone, which is a bit difficult as I don't have a back up the revert to should I restore it to factory settings.

Any help or advice would be greatly appreciate as at the minute I can't do anything with it!

Sam