If your WordPress installation is up-to-date (v3.8.1) and you use the default WordPress editor and your web server meets all the requirements for WP-SimpleViewer, then the SimpleViewer eye icon or 'Add SimpleViewer Gallery' button should be visible in the editor's toolbar.
As you have already tried deactivating all plugins other than WP-SimpleViewer and this did not make a difference, then a plugin conflict is not the cause of your problem.
Please double-check that your WordPress installation is up-to-date, that you are using the default WordPress editor and that your web server meets all the requirements for WP-SimpleViewer.
The previous gallery webpage shows up from 2007 and I am not able to add or edit the existing gallery.
The gallery whose link you posted looks to be a very old version of SimpleViewer and it was not created with WP-SimpleViewer v2.0 or above. It may have been created with WP-SimpleViewer v1.5.4 but there is no backward-compatibility between v2.0 and v1.5.4 (please see here for details) and the gallery whose link you posted will not be able to be edited by the current version of WP-SimpleViewer.
WP-SimpleViewer v126.96.36.199 will be able to edit galleries created with WP-SimpleViewer v2.0 or above only.
SimpleViewer Support Team