Xbox Won’T Switch On

[anaxcrosswords]

My daughter brought her console (the original one) with her when she came to see me yesterday.
A few weeks ago it switched itself off while she was gaming. After a while – don’t know if that means an hour or day – it did boot up again. But not yesterday; no response from the power button and it had been a good few days since it switched itself off. The LED on the separate power unit came on so I know that’s working. It looks like something in the console has overheated and finally given up.
I’ve never had an Xbox, but did have the ‘phat’ PS3 which was notorious for its Yellow Light of Death when the lead-free solder melted. Does the old Xbox similarly have ‘history’ with a particular component failing?
Just want to find out whether the next step is repair or replace.
TIA