Few things you can try first, make sure your outlet is working by plugging another device in.
Next, check your laptop power cord to make sure the two pieces are seriously plug in each other. Also if you got a spare power cord, this is the best option to find out if this is causing the issue.
If everything is fine, put your laptop to charge for 1 hour. It may not display any light as it is charging even if the light is broken but it should not stop your laptop from turning on after sufficient charge.
Now if you press the power button and still nothing, keep on pressing and holding to the power button until you get a reaction. If nothing happens, disconnect your power cord from your outlet or power supply, keep pressing on the starting button for at least 30 seconds to drain the power charge out, then removes the battery from the laptop, if no sign of unusual wear, put it back and try again. Let me know if you got result.