It is quite easy, but some people think there is more to it, than it is:
Once in the Server Room with her, don't be afraid. No Quick time-Stuff will happen.
Walk through the Servers for a Sequence (nothing will happen to you, personally, lol).
After the Sequence go over to the Computer, which will show you the Server Numbers.

You have to activate them in the right Order to go on and get the next Keycard.
Leave the Computer and look a bit up, you can see numbers on each Server-Tower.

Go to each in following order:
25,28,23,35 and then 39.
Each time you have to open a little panel on the right, open it and press "E" at the exact moment, when the Square in the middle of the little yellow Screen is full. Do one by one, inbetween nothing will happen, you are Safe!

After all Servers are online you can walk back to the Computer (next to the Body, where we just have been). A Sequence is played and you will get autromatically the new Level 4 Keycard and load it to your Mini-PC-on-Hand.
Exit now through the Door which says: Atrium.