Area-51 m15x won't power up.

A sorry tale, I realize I may be clutching at straws however maybe someone here can give some advice.
My 16 year old son bought an alienware m15 x laptop on eBay, we knew he was taking a gamble and for about 2 weeks it seemed to work ok.
It then stopped, turned out the filter was blocked and it overheated.
We left it into the local shop and they said the motherboard is gone, it's not economical to repair.
I think he knows his stuff and probably nothing can be done, but I thought maybe one of the techies here could offer some advice.
My understanding is that the machine overheated because one of the "air intake" was clogged with dust. Sorry for using the wrong terms.
The laptop worked fine and suddenly went dead.
I believe they tried new thermal paste, on some part to see if they could get it working, but no luck.

As I'm sure is obvious to you all, I am no computer expert, maybe the shop done all the correct checks, but I would love to know for sure we have done all we can before giving up on this.
I just had a look with my son, the lights on the battery work, however the machine does not power up at all.
The machine is an area 51 model and out of warranty, he bought it 2nd hand on ebay.
he just plugged in the machine, the alien head lit up on the outside and the alien head (switch) lit up on the inside, however only for about 2 seconds and then went dead again.

just to point out the on off button (alien head) does not press down, its more like turning on a ps3, is this normal,

Thanks