How to add a full edit link to front-end editor?

I assume the plugin is a separate thing and does its work by itself.

What you need is to include in your theme template files the default Edit button.

First off, you need a good code editor. Notepad++, Sublime Text, NetBeans, to cite a few.

Check one of the default WordPress themes, TwentyTen, TwentyEleven or TwentyTwelve. You’ll see this: <?php edit_post_link( __( 'Edit', 'twentytwelve' ), '<span class="edit-link">', '</span>' ); ?>. This is the default Edit button that you are searching for. Swap to this theme and try to tweak it until you get the idea.

Probably, you’ll need some CSS to position it near the plugin’s button and to make it visually similar.

Update
Just found this excellent Answer by @MikeSchinkel:
Recommended Books on WordPress for Management and non-Developers?.