Why is WhatsApp and Messenger no longer working on iPhone 5
Because of the closed nature of IOS, there really are no troubleshooting steps that can be taken, besides deleting app/reinstalling, or wiping the device, setting up as 'new' device, and then trying again.

Unfortunately, though the apps list IOS 7+ and 6+ as requirements in the app store, I have seen this behavior on older iPhones (namely my GF's) and the 'fix' has been to update IOS to a newer version.  Though this is not technically a 'fix', as the apps SHOULD work with 8.1.3, it does (for my GF at least) 'solve' the issue.  

Of course, your first step should be deleting the apps and then re-installing them from the app store as I mentioned above.

If that doesn't work, and if you have performed a backup of the phone (either to iCloud or iTunes), you can try simply resetting the phone to factory defaults (keeping IOS at the same version), setting the iPhone up as a 'new' device, and then try to install the apps to see if they work.  If they don't, then you have your answer.  If they DO work, then you can reset the phone once again, restore your backup and hope for the best.

If this STILL doesn't work, you may try to update the IOS version, either via iTunes or the phone's native Over-The-Air update process, but the latter requires there be enough free space on the phone to download the update TO the phone.  Of course, make sure to take a backup first.

At the end of the day, I have personally had to just bit the bullet and start all over, setting the iPhone up as a 'new' device, and NOT restoring from a backup, in order to resolve some issues.  If you elect to take this route, make sure you have your data backed up/saved elsewhere, OUTSIDE of the iPhone backup.  Alternatively, you can use third-party software like PhoneView, iExplorer or iBackupBot to retrieve data from iTunes backups.    

To reset phone to factory defaults (only do this after you have taken a backup): Settings>General>Reset (at the bottom) > Erase All Content and Settings.