I used the core wallet with the full blockchain download, and has since synched with same result. I did notice a suggestion to drop my private keys but not sure how to do this.
First of all exchanges don't give private keys to the users no matter of what, they may help you to resolve which is even happen rarely or else they will let you on your own.
But here the problem is with his wallet so exchange is just an receiving eng which has nothing to do with the sender issue.
be careful to drop the private key, it could be spamming trying to divert your transaction. don't rush. check email messages in all options. if you find a message from Kraken support then I think you need confirmation to make the second deposit or the first deposit. if the tone is experiencing these obstacles, use the support service until you get an answer from them. Again, don't enter the private key.