In addition to @danj's answer, there's a really good explanation that @arbaman posted here with that could give some explanation to the issue. To give you the spark notes of it, it's an issue with the poly-tantalum capacitor. Typically these capacitors go bad over time and will cause a shortage in the board. You can get the device to turn on temporarily by warming up the board, assuming that the computer was in a cooler area, but an actual fix could potentially require a component-level repair if this ends up being the issue.