It's all good. "apt-get install wordpress" makes a mess. We can rule that out.
A couple things you might try just to get a better idea of what's going on. If you want to get a better visual reference of whats going on in the database, you can install phpmyadmin and you should probably be able to access it from the browser on your laptop with no issues.
(sudo or sudo -i, your preference)
you've probably already done this, but just in case:
~# /usr/bin/mysql_secure_installation
Then to install phpMyAdmin
~# apt-get install php5-mcrypt phpmyadmin
after the install, you will need to run
~# php5enmod mcrypt
or else phpmyadmin will squawk about mcrypt not being installed, then run
~# service apache2 restart
If your server is set up with a static private ip, you should probably be able to hit phpmyadmin with your laptop browser from 192.168.x.x/phpmyadmin
See if you can log in as the root user, then see if you can log in as your wordpress database user. That might be revealing if both of those aren't successful.
If you don't want to install phpmyadmin, you might open a putty (or whatever you use to connect) session and log into the server as your regular user account, then see if you can connect to mysql using your wordpress database user.
A successful mysql connection as the user "pete" from my example earlier should look like this:
pete@ubuntu:~$ mysql -u pete -p
Enter password:
Welcome to the MySQL monitor. Commands end with ; or \g.
Your MySQL connection id is 70
Server version: 5.5.43-0ubuntu0.14.04.1 (Ubuntu)
Copyright (c) 2000, 2015, Oracle and/or its affiliates. All rights reserved.
Oracle is a registered trademark of Oracle Corporation and/or its
affiliates. Other names may be trademarks of their respective
owners.
Type 'help;' or '\h' for help. Type '\c' to clear the current input statement.
mysql> show databases;
+--------------------+
| Database |
+--------------------+
| information_schema |
| wordpress |
+--------------------+
2 rows in set (0.00 sec)
mysql>
I hope that might help reveal something database or user related for you.
[edit] .. I think an issue with keys might give you a blank page rather than a database connection message. I haven't verified that, but I don't believe that will cause the database error dialog.