Why was the Inspector laughing/happy at the end? (Spoilers)

I'm thinking that it's not so much him being impressed by how he was tricked, but more the feeling one gets when 2+2 finally come together and he finally sees the entire picture.

He was happy and amused cause he found out it was a trick and they were both alive and happy. Also that kid had the package for him from Eisenheim remember? When he opens it you see it shows how the orange tree trick was done which he'd been wondering about from the beginning. And he didn't really like the prince who actually was a murderer and corrupt power hungry bastard. It's mentioned a couple of times he has killed before like when he threw his ex off a balcony just to cover the beating he'd given her.

I think it's the joy of discovery. They established early on that he was an amateur magician, and he was very intrigued by how Eisenheim performed his illusions.

This was the greatest illusion of all, and he was able to figure it out by himself.

Also, justice was served to the Prince, and the good guy gets the girl in the end. It's clear that he admires Eisenheim and despises the Prince, so, just as in the conclusion of Murder On The Orient Express, it's all good.

Haven't you ever had someone play a trick or a joke on you and even though they were laughing at you, you couldn't help but admire how well it was executed?

It's kind of like that.

He detested the crown prince and was happy to see the illusionist beat him.

Because deep down, he had an admiration for Eisenheim's talents. His only reason to investigate him were due to Leopold's demands and his own wish to move up the ranks through doing so. Throughout the movie, you can see the despicable character that Leopold is and his wish to subject and overthrow his father in order to rule. On top of that, he's a drunk and has no qualms about walking over people to get his way.

Up until the very end of the movie, Uhl has this nagging suspicion that Leopold was behind the entire "murder," but due to Leopold's positioning of power, he can't do anything about it. However, as the evidence becomes overwhelming in pointing towards Leopold as the murderer, he does the one thing that he can do and Leopold's father of the plot to depose of him.

Him laughing and smiling is partly because of him respecting how well the illusion was done, and also shock at how well he was played. Some times in life there's nothing you can do but laugh in the face of defeat and predicament. He got owned.