Should I design my enumeration in some way that indicates what the highest value is?
I think you should add a numeric field indicating the order of a card relative to the other cards. You do not need to use it for anything other than sorting. 

Also, instead of taking the identifier of the enum value and doing uppercase and lowercase tricks with it in order to turn it into something presentable to the user, you should just store the name, too, as a separate string in the enum. 

So, it would look like this: ACE( "A", "Ace", 0 ) (where 0 indicates lowest order for sorting.)

Furthermore you can also add one more number, the 'weight' of a card, to use in calculations which determine whether a card beats another. In general, enums in java are very powerful, so powerful that you pretty much end up not having to use the switch statement with enums anymore, because you can include a big part of an enum's functionality within the enum itself.