As far as I know, it was never stated that you must have wind affinity in order to complete Rasengan. Combining Rasengan with nature element is extremelly hard, that's why Kakashi or Jiraya did not complete it. Naruto did it because of his special Kage Bunshin training, which allows to learn a new technique in no time. Otherwise, he wouldn't master it yet. It could take him years to do it.The only person to Master the rasengan is Naruto.
In order to master the rasengan you have to add nature manipulation into it.
Naruto did this by adding his wind nature to the rasengan to make the Rasenshuriken.
Jiraiya did not complete this technique because his affinity was not compatible to further the rasengan.
Kakashi could not complete it because he does not have wind nature and wind nature is needed to be compatible with the swirling motion of the rasengan.
Minato did not complete it but did know it was incomplete but maybe died before being able to finish it.
-Naruto is the only one to Complete the rasengan!
And for those saying, that Kakashi copied Rasengan from Naruto, it is not true. He learned a long time ago. He created Chidori only because, he couldn't combine his nature chakra with Rasengan. (Chapter 321 Page 10)