Okay, not sure how much you know so will start with the basic outline.....Im sure someone else will jump in with more info too
to create a collection in the main library you need to highlight LIBRARY in the left pane, then click the + sign at the bottom left......
First try this though as it should solve the issue.....
With the reader connected via USB and switched on, open the eLibrary. In the left pane you will see the reader show up...........click the triangle next to it to expand the info for it. This should show you what collections are on there and synced ones have the sync symbol on them. To delete one click the sync symbol in the reader info. This should show you a list of collections in the right pane. 'Jeff' one should be there with a tick in the box. Untick the box and that should empty it on next sync then allowing you to delete it from the left pane by right clicking on the collection 'Jeff'..........
let us know if this works...it should do as it will show all collections on the reader............and allow you to un-sync the 'Jeff' one allowing you to then delete it.
I sort my collections by adding them to the main library then clicking the sync symbol on the reader in the left pane and ticking the boxes I want synced on the reader. Unticking the boxes will allow emptying and deletion of collections. As you have found, while a collection is synced you cannot delete it. Very odd that the 'Jeff' collection isn't in the library section though, that one has me stumped.