I have a horrible feeling that you will need to see the original display first, to set the options for the external monitor, which of course has you in a vicious circle in this case, I don't really know those type systems, but I know on mine, I MUST go into the display options and set the system up for an external display, as there is more than 1 type of setting to do this, at least in my case, others used to multi display setups may have some better advice, but I have a horrible feeling you will need to repair the original screen before you can do more, I hope I'm wrong for you, but it's what I suspect will be the case.
Now as you say support does seem limited, and I have only been able to find 2 bit's of advice to do with this, 1 of which is hopeless, as it requires you accessing the bios to change a setting, which of course you can't see, or the following which is for a laptop, so I have no idea if it will be of any use, but given the situation I figure it can't hurt to try it, so see if the following is of any use:-
2. Ensure the system is not in suspending mode by pressing and holding the Fn key for 10 seconds.
3. Toggle between the internal LCD display and an external display by pressing the Fn and F3keys simultaneously on the notebook keyboard.
*There are generally four display mode settings to toggle through.
Step 1 is about power so not needed, step 2 I doubt will do anything, but step 3 is the only possible hope I have been able to find, now there are 2 important things with what you are trying to do, which is made critical by the fact you can't see anything, first is using the correct output, the HDMI would seem to be what to use, I would check your external monitor to see it has the right input connection for this, some connections can be output only, which will not work for this.
The next important factor is to ensure the external source is set to the correct input, eg if you connect to say the HDMI 2 connection, the display must be set to the the HDMI 2 channel if that makes sense.
I know many have auto sensing\switching ability, but when you're working blind you have no idea if thats working or not, so making sure it's on the right channel is 1 less thing to worry about.
Finally is the note of more than 1 type of display mode, when switching between each mode it often takes a few seconds before it switches through and you see the other display kick in, so give each attempt a few seconds to work, then it is having the right one that will come through, the modes go from "Computer only; Duplicate; Extend and Projector only" so I figure it's the 2nd and\or 3rd ones that you would need, this assumes this actually works, not sure what else to suggest, hope this does work for you.