Boost circuit for LCD matrix display

The reference in the manual to a 15 pin D-sub connector relates to the provision of video on a socket on the outside of the laptop for connection of an external display. This has nothing to do with the internal interface to the internal LCD.

I suspect that the easiest "fix" is to decide how to get your external display to fully mirror the internal (dead ) one and then produce / acquire the simple program required to set this up properly both at turn on and any time you enter a selected key sequence.

TGG has mention what appears to be an HDMI connector in your LCD photo. If this is what it is and if it intended for video interfacing to the display it will be the easiest way of achieving what you want to do.

There is no certainty at all that the interface to your LCD is DVI - it probably isn't. If the laptop is 'somewhat old'. that MAY be a VGA connector.

LCD displays have used DVI, VGA and many use

Discussion here on what Dell Inspiron uses.

Parlez vous Francais?
This page is in French They claim to have achieved a working connection to a laptop LCD display. Wrong direction for you and only relates to that model or similar but useful.

This page says

• LVDS is the signaling system used with most flat panels in laptops. FPD-Links (Flat Panel Display Links) are the actual interface (like DVI) that most laptops use to connect LCD panels to graphics boards. FPD uses LVDS.

  Another interface used in modern laptops is called OpenLDI which also uses LVDS signaling.

You don't want to know :-).

Do you have a diode across your relay? If not the inductive spikes on closing/opening will likely cause problems.

When you e.g. stop the current flowing through a relay, it tries to keep it going, if there is no route of discharge it will create a large voltage (essentially it will keep rising until it finds a route).
Specifically the formula is V = L(dI/dT). This means the inductor opposes changes in current through it by developing a voltage across it proportional to the rate of change of current.

Try placing a diode across the relay, orientated to oppose normal current flow.

EDIT - checking the product page it seems it is a "ready rolled" relay in a box with peripheral components added, so it's unlikely to be the above. Even so I would try placing the diode across the power to the relay.
Also make sure your supply is well filtered as mentioned in the comments. Place a few capacitors of at least 1uF next to the ICs and LCD, and a bulk cap of >100uF somewhere (all from power to ground)