Are you sure it isn't actually on and just the display isn't showing? If that is the case, try option 1 below. If it isn't turning at all (no lights, etc.) then try option 2.
1. Turn the laptop off (not sleep or hibernate but off).
2. Connect an external monitor to the laptop.
3. Turn on the external monitor.
4. Turn on the laptop.
NOTE: You may have to press an "external monitor" button. Could be the f4 button or a button with two monitors on it, for the external monitor to work.
If you can see fine on the external monitor, then your attached display, or the ribbon cable that connects it, are your problem.
If you can't see on the external monitor at all, or the problem occurs on the external monitor as well, then it is probably the graphics card/GPU that is the problem, which may require the motherboard be replaced.
Try the following and see if it will turn on...
1. Unplug your charger cord.
2. Remove the battery.
3. Press and hold the "Power" button for 30 seconds.
4. Now plug back in the power/charger cord.
5. Now put the battery back in.
If this doesn't help, then test the power cord to see if it is good. Either try your cord in another device, or try another devices cord in your laptop. You may also want to try charging it when the device is off, rather than on. If it works then, the cord is probably the issue.
Should that not be the problem, then check the battery to see if it needs replacing.
NOTE: While the battery may be complicated to get to, it should be replaceable. Unless it is under warranty, then you should contact the manufacturer.