This is not an issue with the lid switch, either your motherboard is fried or the power button is broken. Having a closed lid would still enable the laptop to boot up, but you would not get a display. If you hit the power button and you get no drive activity light, or anything other than batter and plug lights, you have a hardware issue but not with the lid.
Lid switch does a few things: close it, it suspends or hibernates the laptop, or swaps the video output to the external connection, or nothing if you have it configured for that. If the laptop is closed and in a dock, if your lid is closed it automatically moves video out to the external display, but it does not prevent the laptop from turning on.