Use of Monitor Output Socket

If there is any way you can find another power supply that you know works and try it, this will be a really good troubleshooting step. If you know anyone anywhere that has a similar laptop. Or try your power supply on another computer that takes a similar power supply, to see if it works on a known good laptop. Also, try using your power supply on your laptop with the battery removed. If the battery is bad enough, it might make the laptop not boot, even though it is plugged in. If there is no other laptop or power supply available, you should at least be able to use a multimeter to test the output voltage of the power supply. The expected output voltage should be printed on the power supply.
If the plug is loose in the socket on the laptop, the socket may be broken off the motherboard. Some laptops do not have the socket soldered directly to the motherboard, but most do, requiring a motherboard replacement if the socket is broken off.

When you connect the power adapter into the laptop, does the power LED light up? If not then -
The battery adapter or the adapter's power lead or the laptop's power socket could be faulty.
Check power adapter’s DC plug that connects into the laptop power socket with a multi-meter, the voltage should be slightly higher than the voltage that is printed on the label on the bottom of the adapter. If the voltage is zero or way below the voltage that is printed on the label then the power adapter is faulty and needs to be replaced.
If the adapter is OK then connect it to the laptop and power it up, then look at the power/battery charging LED light and wriggle the power plug in the laptop's power socket. If the power/battery LED light flashes and in a certain position this LED light stays firm, then the power socket is either faulty or it has a dry solder joint where it is connected to the motherboard.
This can be repaired but it requires the laptop to be completely dismantled to get at this power socket and repair. If you cannot DIY then I suggest you get a quote first.

Look at the power/battery charging LED light and wriggle the power plug in the laptop's power socket. If the power/battery LED light flashes and in a certain position this LED light stays firm, then the power socket is either faulty or it has a dry solder joint where it is connected to the motherboard.
This can be repaired but it requires the laptop to be completely dismantled to get at this power socket and repair. If you cannot DIY then I suggest you get a quote first.

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There should be an audio cable coming out of Monitor, you'll need to plug that cable in the ouput jack of sound card, turn on/increase the volume of the speakers using the switch, slider present on the monitor and then check. you may also consult the manual that came along with the monitor.

you should have a (vga) tv-out socket usually blue with about 16 pins in it,if you have not which is unusual then refer to sony for the laptops specs and location of tv-out socket.
The Nvidia graphics card should have one I believe the Gforce go card is the one for your system unless upgraded and is advertised to support this function.

Some monitors have speakers built-in, some don't. The sound never travels through the 15pin d-type VGA connector. If the monitor has sound, there will be a small 1/8 inch stereo jack in the back or bottom of the monitor. Most all laptops have a headphone output jack (again 1/8 inch stereo) which is perfect for patching to external amplified speakers or to a monitor with speakers built-in. This patch cable can be purchased at Radio Shack, Best Buy, etc etc.

check adapter output also may be problem with dc socket on laptop loose or somthing.