I think if the Kobo icon is shown in the pane by ADE then it's authorized, but you said you had some other problem, so you may want to de-authorize the reader and then re-authorize it.
With ADE open and the reader connected press Ctrl + Shift + E. ADE will ask if you are sure and you select Yes.
Then unplug and then replug the reader into the PC and ADE will popup a box saying a new device is detected to be authorized.
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