If you mean the Epic panel to the left, it displays all issues of type Epic visible as part of the filter that you used for the board. The drop down menu for the Epic does not have a Delete option. You need to open the Epic in JIRA and delete it. Which documentation are you referring to?
EDIT: Can you get us a screenshot?
Waitl Closing an Epic has no effect on GH. GH uses a custom field 'Epic Status' which can be changed only if you have Edit Permission. That is why I asked you to try Editing the issue. Can you confirm that you are able to click the Edit button while viewing the Epic issue directly in JIRA?
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