Does the laptop have an D sub vga output? If so does this work?
If you boot the laptop with the HDMI connected do you get any thing at all on the screen as it boots?
It is relatively easy to pop the laptop HDD out and connected via a usb adapter to another PC and copy the contents.
Another option is to look for a local repair shop and get a quote for a screen replacement. ( I got quoted £115 for a 15" Dell replacement, less if I wanted a 2nd user screen.)