The approach that i would take is
Tell your line manager that the server is running out of space and that we have x amount of time to do some thing about. Make sure that other people know about it as well if you can his line manager as well.
Then tell him the only way to extend the life of the server is that stuff needs deleting and that All members of staff need to delete work (this is now creating work for him and other members of staff). The response that you will probably get is "can you sort that out" and your answer is NO as you do not know what is important to staff.
Make sure everything is in E-Mails and cc the Head or the Bursar as well to cover your own back. Let him stew for a couple of days.
Technician to the rescue time
If we got a second server in and added it to the domain it would give us
Additional disk space
A second DC to act as a back up controller WHEN THE OLD SERVER FAILS not if
If you get the new server promote it to the master controller then keep the old server as a secondary.