If you boot up the laptop with the external monitor already connected, do you see the BIOS screen (hit "F2 to enter setup", or something similar) on the external monitor? If so, you may be able to change a setting in the BIOS that forces the external monitor to be switched on, when you first boot. If you can't see anything on the external monitor, you may have to do that step "blind" - if there's the slightest chance you can get an HP DV4 owner to work out the necessary steps...
Replacing the LCD screen: around £50 for a screen on eBay, typically. Further £50 to get someone reputable to replace it.