Our server admin noticed that an External HDD we were using for backups was not showing up and Windows was giving an error that the device could not be installed. He unplugged the drive and restarted (again), and now the server is running MUCH better. Database Cache Memory is now over 20GB and disk IO for the server is averaging 5MB/sec and below.

He said that we had a problem with some of our file servers last year with external drives (USB 3.0 drives plugged into a USB 2.0 port) with very similar symptoms - the servers ran very sluggish with no noticeable reason.

I have no idea how an external hard drive could cause memory issues like this, and it is possible that the fix was related more to restarting again. However, at this point all I can do is wait and see if it happens again.