Here is a simple solution for getting your contacts at the very least. This is what I did from one iPhone to another iPhone.
Login to the iCloud account associated with the device you are trying to retrieve this data from.
Once you are logged in click on the contacts tab in iCloud. In the bottom left hand corner you will see a "setting gear". Click this and then click select all.
Once all have been selected, click the "settings gear" and click export vcard. Save as contacts somewhere you will be able to easily locate like your desktop.
Log out of that account and log into the account you wish to transfer this data to. Once you are logged in to this account, click contacts and the "settings gear" in the left hand corner. Click import vcard, locate and select vcard imported from other account. You should now see your contacts transferred from one account to the next.
You will then need to click the "setting gear" again and click refresh contacts. Power off your device associated with the device you want contacts transferred and power back on.
Once everything is updated you should see your contacts on your new device.
Best of luck!