There are two types of admin on JIRA - System admin and JIRA Admin. A system Admin can do everything. A JIRA Admin is restricted to looking after the application. JIRA Admins can add you to their group, but only system admins can add you to system admin groups.
You are on Cloud, so you will not get system admin anyway.
But, you only need JIRA Admin to delete a project, not system admin. Go to the list of projects in the admin section (*not* the users list you get when you click project in the menu) and you'll have a delete option to the right of every project.
The default admin user on the internal directory should have system admin rights, so you could temporarily log in as this user and set your user in the jira-sys-admin group. Somebody in your org, like your IT sys admin, must have these permissions as well.
That being said, permissions associated with this group are the highest possible in JIRA. Very few people should have them. If you're not the main admin in your JIRA instance, perhaps it's for the best that you don't have these permissions. Either way, you'll need to contact somebody with sys-admin rights.
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