You don't defeat the investigator -- Black Widow, an art thief, is posing as the investigator to steal The Scream. When you find the painting in the investigator's cottage, she captures you and takes you to her underground lair.
(see the related question)
The key to the investigator's chest on Counterfeit Island is hidden in the bushes to the right of the cottage. You need to use a stick to reach it and then unlock the investigator's chest to get the next clue.
night right after you get caught the thief is the investigator
it's the little cabin at the very end of the island, if you need help check your map :)
The investigator's house is all the way right (as far as you can go).
The investigator is another of the terms used for the character of the lady Chief Inspector. She is at the museum when you arrive, and arrests you that night. Fortunately, she needs your help.
You beat her. She's in prison.
Presumably off in prison somewhere. She is not in the jail.
Go inside the house and peel the painting off the one on the top floor
in the expressionism part of the museum
You will not see the investigator anywhere after you finish the security tapes in the museum. However, now that you have the key to her house (in the Country Side), you should find something there.
The lady Chief Inspector is nowhere to be found once you leave the Security Room with the photo of the art thief.
Jump on the light hanging above the Scream to catch the theif and wait for the "investigator" with the police.