You can have a global integration or integration at the project level. Sounds like the one project has integration set up and the rest do not. That would say that the integration for JIRA was not at the Global level.
In my works instance we integrate with TFS and when I click on the Integration portion of the Admin screen I see the notice below at the top of the screen:
This is the first thing I would look at since it works in one project (so integration is set up with the correct values) but the other projects integration is not working.