When you get right down to it...

Hummm....that wouldn't make much of a movie would it???? :)

OK, maybe none were available.

"Some are born to fail, others have failure thrust upon them."

It's mentioned in the Scotland Yard scene that the diamonds are too famous and known to be sold unrecognized. Gregory needed to become official and legal owner of them, then he could profit off them. That's why he first needed to marry Paula and then get her declared insane, thus gaining the custody over her and her property.

I believe it was mentioned in the film that the house was owned by Paula all that time after the murder - why else would all of her aunt's possessions be kept there? And it would be hard for Boyer to do rooftop moonlighting if all the possessions hadn't been moved to the attic first... as they had been after he and Paula moved in.

Gregory wanted a legit way to search the house. If he broke into an empty house and was heard/seen by the neighbors, he would have been arrested. Also, who knows if Alice Alquist would have even kept the jewels in her home? Or if she did, who is to say that her niece or someone else did not move them elsewhere? In order for Gregory to be sure that he could find the jewels, he needed to get close to the one woman who could tell him if all of Alice Alquist's things had remained in her house (or who might have revealed that yes, she inherited jewels from her aunt). One also wonders when exactly Gregory came up with his plan. Had he been planning to woo Alice Alquist's niece for years, or did he simply come across Paula and decide on a plan then?

Also, I get the feeling that Gregory did not want to sell the jewels. The scene at the Tower of London with the crown jewels gives off the distinct impression that Gregory is obsessed with gems in and of themselves ("They have a life of their own!"). My read is that he simply wants to possess jewels for the sake of possession as one might want to own an original Van Gogh or a trophy wife.