From what I've read, New Game+ carries over your health, void, and chaos meters, and gives you all of your experience and mastery back so you can buy all of your old skills again, or if you beat the game without all of them bought, buy different ones first this time around.
There is a separate trophy/achievement for completing all of the Kleidos on Prince of Darkness mode specifically. So either they don't carry over, OR the Kleidos Challenges are unique to that difficulty with different enemy sets and optional conditions.
Either way I'm sure beating the game on Prince of Darkness counts in and of itself.