These are great opinions; Sunarep's idea of Joker as a "Dr. Lecter"-type character is especially potent. However, TM2YC's statement makes the truth obvious - we have no idea how TDKR would have been had Ledger still have been alive (short of asking Nolan himself).
Personally, I think that if that were the case, and if it were the same TDKR story, then perhaps Joker would have been kept nice and safe in the worst dungeon of Arkham until Bane invades and personally frees him (a little DC villain fanservice). However being the Joker, he would have just wandered within the limits of Gotham dealing the usual chaos for the fun of it, especially if he knew they were doomed, anyway (perhaps a Scene with Catwoman defending that little kid from him instead of a simple goon).
Finally, with the city saved and Mr. Bad Throw-In Character Blake inheriting the Bat-mantle, the last teasing hint is of Joker at large again, and any future plot would involve Blake's (as Batman) eternal war with Joker, as a last tribute to the canon comic's relationship. Either all this, or Nolan's realism just forces Joker to rot in Arkham and make up for comic! Joker's absurd avoidance of justice.
Would anyone think this a good alternate plot for Dark Knight 3? Perhaps a possible edit?