That's why I always tell people;
1) Install the Battery
2) Connect the AC adapter (Charger)
3) THEN connect the AC adapter to power.
AC adapter disconnected, remove the Battery.
Press the Power On button in, and hold it in for a count of 10 seconds.
(And I mean 10. Count, 1001,..1002,..1003,..1004, etc)
Let go of the Power On button.
Do this procedure 2 more times.
What we are doing here is clearing CMOS Error Codes, resetting BIOS setup back to the factory default settings; and resetting sensor values in the power circuit section of the motherboard.
Now reinstall the Battery. Reinstall the AC adapter. Plug back into power. WAIT 1 minute, turn the laptop on.
Suggest we have a problem Houston.
Power MOSFET's in the power circuit section of the motherboard, or may be just a bad Power On switch, now.
The Sony VAIO PCG-7184L is also the VAIO VGN-NW235Fhttp://www.insidemylaptop.com/replace-lcd-screen-sony-vaio-nw235f-laptop/http://www.youtube.com/watch?v=F3KQ6_SlJsw
Use an ESD wrist strap, and have it's alligator clip attached to a good ground source. Average cost is around $3 to $6.
I connect to an unpainted surface, of the metal frame of an open, empty desktop computer case.
I feel an Anti-Static Mat is unwarranted.
You can also set a large metal serving tray (Unpainted), on the table you're working on, and connect to it.
Or a large metal knickknack. (Unpainted)
Use a multi-compartment container for the various screws you will remove.
Label each compartment for the area the screw/s come out of.
SOME screws look VERY similar to other screws.
Advise DO NOT mix them up.
Look at the motherboard where the DC Power Jack is mounted.
(The DC Power Jack is where the AC adapter plugs into, on the laptop)
You'll see Power MOSFET's, and MLCC's surrounding them.
[Multi-Layer Ceramic Capacitor. They are also SMD/SMT, as the Power MOSFET's are.
Surface Mount Device/ Surface Mount Technology ]
This is an example using HP Pavilion dv6000 series, and Pavilion dv9000 series, of Notebook PC's,http://mayohardware.blogspot.com/2010/04/important-parts-on-dv6000-and-dv9000.html
Uses a Fairchild Semiconductor International - FDS6679 - Power MOSFET, and an Alpha & Omega Limited - AO4407 - Power MOSFET.
Looking at the second photo down with the AO4407 circled in Yellow;
Look at the rectangular smaller object to the immediate Right; that has the dark wide band across the middle of it.
This is an MLCC.
There is one above it, one above that one to the Top; one to the right of the top one; and one to the right of it.
See if any of these are burned/blistered also.
Or it might be worse, Clifton,http://koti.kapsi.fi/jpa/stuff/pix/travelmate_6292/
May be time for motherboard replacement.http://www.ebay.com/itm/Sony-Vaio-VGN-NW242F-PCG-7184L-A1747083A-3276432-Laptop-Motherboard-Tested-/281040027261?pt=Motherboards&hash=item416f4a7e7d
Not advertising for seller, nor website. Just presenting an example.
For additional questions please post in a Comment.
[More examples of procuring Power MOSFET's, MLCC's, and other related electronic components; plus procedures for de-soldering/soldering SMD/SMT components,
(Use a low wattage soldering iron (25 Watts?), Rosin core solder, rosin flux paste, and flux paste brush; instead of a Soldering Station.
Use Isopropyl Alcohol, and an old toothbrush to clean the motherboard, after removing components, to remove the residue left behind.
Isopropyl alcohol (Rubbing alcohol), is EXTREMELY FLAMMABLE.
Use in a WELL ventilated area with NO sparks or flames present )http://www.mouser.com/Search/Refine.aspx?Keyword=FDS6679
One dollar, one penny, USDhttp://www.mouser.com/Search/Refine.aspx?Keyword=AO4407
AO4407 is phased out. Using one of the cross-reference examples given,http://www.mouser.com/ProductDetail/Diodes-Inc/DMP3020LSS-13/?qs=%2fha2pyFadujguNNSgppBRqXbOxXLwzGsZm9SWxz1EqWpzoSJOpHpWA%3d%3d
78 Cents USD.
[ The 'legs' on the sides are J-leads ]
MLCC's that are SMD/SMT, and 4.7uf (MicroFarad) examples shown, at 16 Volts (DC),http://www.mouser.com/Passive-Components/Capacitors/Ceramic-Capacitors/Multilayer-Ceramic-Capacitors-MLCC-SMD-SMT/_/N-b2cj?P=1z0wrjcZ1z0x565http://www.youtube.com/watch?v=5uiroWBkdFYhttp://www.youtube.com/watch?v=b9FC9fAlfQE