Will the computer turn on if you plug it into another known working outlet?
If not there was probably some sort of short circuit created when you attempted the plug and and electrical current went the wrong way into your computer.
First have someone knowledgeable check the uncovered outlet, fix it, and get a cover plate on. Would normally hope that whatever happened would have just tripped the circuit breaker (if there even was one). But the lights being on indicate that there is some level of power at the outlet and that the outlet is live. And can shock again.
Second, with your computer unplugged open it up and inspect all components for damage: black or brown scorch marks, melted wires, melted components. any burning smells...