The issue here is that you need to set it up as a secondary monitor but can't because the main screen is bad.
Try hooking everything up, turn on the laptop and quickly close the lid. You'll need a keyboard and mouse attached of course to use the computer. The laptop should detect the external monitor as a main screen.
Another option is to see if your laptop has a docking station you can get, but most home models don't have one that works properly. You need one with a power button so you can just keep the laptop closed and turn it on with the dock.