A little more information about your setup would help us to try and help you.
How are, or were, the drives setup on your DNS-320 - RAID, stand alone etc.
What operating system are you using?
Where and how did you install your new printer?
What do the indicator lights show on the front of the DNS-320?
Since you can access the NAS via the web interface, it does not sound as if installing the printer drivers on your computer is really he problem unless those drivers changed the subnet of your router - something they shouldn't do byt can if you do not understand the available settings that networked brother printers have.
I would therefore check that the router is 192.168.0.xxx (xxx being its default - lookup the manual) and not 192.168.1.

- to which the brother setup can change it.
If you are using windows, get something like Netscan to find out what IPs the various parts of your network have been assigned - the DNS-320 should appear there as well as your router.
If you actually have a problem with the disks of your NAS and you have no backup there MAY be a way of recovering your data provided they were not setup as RAID 0.
Another point, if you have removed the disks make sure you have labeled which slot thet were in and be sure you return them to the correct slot.