Does display work through an external monitor? Can you press the switch and turn it on?
If all those don't work, does it have an Nvidia chipset or Graphics in it? HP's and Nvidia has a huge problem with lead-free solder and their chips, so much so, there was a class action lawsuit against it.
Tried an external, no joy. It's all Intel, so I'm a bit surprised that the GPU wouild fail, especially as it was turning itself off after a few minutes according to the owner. Should I spend another £3.50 on another switch...