How about something like this?
Used a free puzzle generator and imported that svg into inkscape and them applied the Whirl extension to it (extensions-modify path-whirl). The outside line did not move at the same rate as the inside pieces so i broke the object apart and deleted that and added a quick stroke to outside rounded the corners and then hit ctrl+L to smooth it out a bit.
Edit: made this one from the same puzzle but broke it app art first then tweaked the dimensions of the outer box to kinda match your shape above. then made the inner puzzle section smaller and inside that object and used the envelope extension.
