Hi!
Well, the most common reason for an Xbox Original that won't turn on, are the capacitors on the motherboard it self.
Next to the power supply, there are 3 (or 5, depends on motherboard model) 3300uf 6.3 V capacitors.
Check those, those are likely to fail and go bad and need to be replaced.
Reference picture: [link|https://retroconsolemods.ca/xboxcaps|XBOX Capacitor Repair|new_window=true]
If possible, please share some pictures regarding your motherboard, so we can check as well.
Thanks!