How do I add that read-only submission date field to my existing form?

In my demo form http://form.jotformpro.com/form/52557605632962? that was just the DateTime field (from Quick Tools)

I have set the Read-Only to Yes in the field properties.

You may want to clone the form also using the URL http://form.jotformpro.com/form/52557605632962

 so that you can view it on your Form Builder.

user guide: -How-to-Clone-an-Existing-Form-from-a-URL

Let us know if you need further assistance.