You have two options:
1) if you get access to the Dropbox account, you might recover the files through the web-interface.
2) If your ex employee would not provide the log-in info. Then you can recover the data even if they are deleted and new data is overwritten on the old ones. However, the digital forensics may not be easy if you are new on it. Then I would strongly suggest you to contact with companies that does data-recovery, I think it would be the easiest option so that they can recover all lost file as they use good tools. It is even possible to recover the data with a professional tools even your hard disk is physically crashed. If you do Google search, you should see the available companies as Google provides result (or ads) considering your location.
In any case, do not use your computer as new data might be overwriten on the lost ones which will make harder to recover the lost data. If you want to continue using your computer, then take image of your hard-disk, so that you can continue using your desktop or if you try to recover the data by yourself, you can attempt to recover data from the image file which will ensure that your PC's file system will not be damaged . You can use Winhex to take the image and it also provides data-recovery options. but it is a basic program. More professional (and commercial) tools would be EnCase, Forensic Toolkit, DMZ F.I.R.E., Maresware
But as I said it would save you from a lot trouble if you contact with a company which does data-recovery.