In my opinion there is a fine line between not leaving the success panel up with enough time to allow the user to read the message before auto-hiding it - and leaving it up so long that the user starts to wonder if and how they need to close it themselves.
Since that line will vary from person to person, you are simply not going to cater for all people all of the time and you are therefore only going to be able to cater for an 'average' user - whatever that is for your particular audience.
The variation in what the user finds 'ideal' is a range, but your chosen delay period has to be precise.
In other words - there is no one-size-fits-all. For this reason, to avoid the problem as much as possible, the success box needs to contain as little as possible, and not contain information of value which is lost once the box disappears.
After the user has subscribed you could change the green/white icon in the trigger area to be a question mark or similar, and change the word Subscribe to Your subscription or Subscription information or similar, so that the user can find out about the subscription that they just subscribed to - or yesterday or whenever it was (in the case of return visits).
After all - they're not going to want subscribe again - but they might very well want to access the subscription information (which email was used, option to unsubscribe, as well as access to the link that was in the success panel before it slid away. Otherwise the trigger area becomes redundant after subscription anyway.
If you switched the icon and label as the panel slid away, user's might notice the change more - you could even fade the panel as it slid away, so that the changes on the trigger area were then more noticeable - user will be looking in the general area as the animation happens but you don't want to bury the changes in the trigger area with the distraction of watching the animation.
This would all then help to alleviate the disappearance of the panel because the information is not not lost in the process.
You could even rethink the strategy as a whole and remove the success box completely - in favour of a simple success/thank you message in the trigger area, which then turns into a 'Your subscription' button as above.