I am trying to troubleshoot a laptop with LED screen backlight

The LEDs are SMD LEDs on a thin strip, arranged in what looks like several 3-LED series circuits, connected in parallel to the output of the driver, no resistors. The driver is boost converter (coil, chip, and few passive components) running on DC

The laptop fell on the floor, and the backlight stopped working completely. The owner tried to repair it (and broke the connector between the LED strip and the driver in the process), then now is my turn to try to repair it

I removed the broken connector and soldered the wires directly - the LEDs work dimly and flicker randomly (similar randomness to as if there would be arcing connection somewhere in series with a high voltage powered light)

Pushing on the inverter components with plastic object does not affect it, but if i touch (so close with high resistance) the controller chip then the inverter goes out completely

What can be wrong with it ?

I know that the driver was powered with its output open for a while, would it damage it in this way ?

Maybe it was run momentarily with the output shorted, would it damage ?

What to look for ?

Pushing on the inverter components with plastic object does not affect it, but if i touch (so close with high resistance) the controller chip then the inverter goes out completely

The prime cause was some broken connection. Either to the LED, or to the main board of the computer.

I know that the driver was powered with its output open for a while, would it damage it in this way ?

It depend, if the circuit have implemented an OCV limiter. If not, the driver chip, diode and/or or the output capacitor may get damaged. If the broken connection is intermittent, the capacitor may get charged and then discharged to the LED's, damaging  these.

Maybe it was run momentarily with the output shorted, would it damage ?

If you look to boost converter basics, you would see, then the short n the output practically mean short of the input, without the controller ability to influence the current (even if you disconnect it completely, the curret would be there). In practical LED circuit that mean huge current flowing through at least the coil, diode and current sensing resistor. What is usually the most sensitive is the current sense resistor - normally is on it the voltage drop below 1V, but during the short circuit there is nearly the full input, what usually blow it out.
These usually fail open circuit, causing with nearly no real current the controller "think" there is already too much, so shutting down.
As in the normal high frequency cycle the booster deliver the energy after the switch turn OFF, it cause some minimum amount of the energy is transferred to the LED, once the converter is fired, so the flashing. When you touch that circuit, you most likely feed the feedback circuit from the input, so the controller "think" there is too much LED current, so keep shut down.

What to look for ?

First look, what chip is there, look into it's datasheet on how it work and identify all related components on the board.

Identify all inputs of the boost IC basic circuit from the rest and fake them in order to let it light at full power.
Then take a scope and look for waveforms.

I replaced all wiring and all the passive components except the coil. This had no effect

So i think that its the controller chip - which i could not identify

It is SSOT6 package and seems to be same pinout as Fairchild FAN5331 (by what component is connected where) but not the same - There are 6 pins in the chip and the extra 1 is connected directly to Vout. And i think that it outputs way more than 1W of FAN5331

When i disturbed it (trying to get some epoxy off it) the package cracked - so perhaps it was exploded from overheating (smoke released) before i got it

Meanwhile the user decided to buy a new LCD panel alltogether, but if he won;t take his old one maybe i'll get to experiment with it some more

If the IC exploded, there is nothing much to discuss... :-D
The question is, whether there is some overvoltage protection - e.g. a Zener between the OUT and FB (having one exra resistor to the Rsense, so the Zener does not take as much current).
Maybe it is the 6'th lead, it's connection look like...

And then other mechanism came to my mind: As the NB fell, it may damage the wiring to LED's so, the wires short together. That then mean har output short with all of the consequences I described before, plus the root cause fault may be still present after you manage to repair the driver.

For the output power it strongly depend, what is the input voltage. The FAN5331 have 1A switch, what could mean more then 3W with 2-cell LiOn battery for the power input.
Don't forget, then the IC may be supplied (pin 5) from different (lower; e.g. regulated 5V) supply line then the power input (the left side of the coil), allowing higher voltage then 5V on the power input, while still stay within IC limits, thus increasing the transferred power.
It then depend on the coil inductance, bt when it is the larger one, he power may go nearly to 5W at 6V input...