How can I reset the admin password without a working screen?
Simply create a new admin account by reinitiating the OS X Setup Assistant: 

Boot to Single User Mode by holding ⌘S on startup with the external display attached.

Wait a few minutes and then enter ("blindly"):

mount -uw /


wait a second or two and then enter:

rm /var/db/.AppleSetupDone
reboot


In Single User Mode the US-en keyboard layout is used. If you have a non-Apple or a non-US-en Apple keyboard you have to adapt the input.

After the restart the OS X Setup Assistant opens and you can configure your Mac. Go through the steps of creating a new admin account.

After logging in on the new admin account, go to the Users &amp; Groups preference pane. Select the old account(s), press the minus button for each and delete all files associated with them.