You want to have the Standard Edition Front-End server located on your internal network and a member of the internal AD forest, then deploy a second server in the DMZ, either stand-alone or in a seperate DMZ forest, to host the Edge Server.
Since AD is required I'm assuming you have an AD forest in the DMZ where you deployed the schema/forest/domain updates for the install? You'll need to uninstall everything and start fresh inside the perimeter firewalls.
Also, the Edge server by itself can't exist without the internal Front-End server.