When you meet Cleopatra she will ask for the Hope Diamond as a tribute.

Go back upstairs, past the gate and use the Animal Power on the rhino to kill the guy. Enter the ice area and use Animal Power on the polar bear to tame him. Then use your Paintbrush power on the penguin painting and press X to enter it.

Use Animal Power on the polar bear to knock over both ice columns then use it again on the rhino to break down the gate. Go into next area use the polar bear to knock down another three ice columns. You can now use the rhino to break down the remaining two gates.

Use your Animal Power on the Saber Tooth Tiger (which should give you "The Light Touch" achievement) so you can grapple on to the top of the case with the two guys. Press X for a cut scene and you'll receive the Hope Diamond. Return this back downstairs to Cleopatra for the achievement