Lcd goes dark after 15 minutes

Try attaching an external monitor to the laptop & see if you could get any display on it. If this is successful, then you might have either a faulty LCD cable or screen inverter board.
The LCD cable transfers data signal from the motherboard & video card to the LCD screen. It also supplies high voltage DC power to the screen inverter board. The screen inverter board works as a power supply for the backlight lamp mounted inside the LCD screen. It converts low voltage DC power to high voltage AC power needed for the backlight lamp.

Its not the inverter, it's your backlight. Hard work to change, sorry, and it might be cheaper/quicker to replace the whole lcd panel at $100 (£65)

Have you tested it with an external monitor?

I would say you either have a damaged inverter board or screen baclight lamp. Usually the inverter board fails in one of the following ways:

1. When you turn on the laptop, the screen lights up for a short period of time and then goes dark. You still can see an image but it is very dark, almost invisible and definitely not usable. When you restart the laptop, the screen lights up and then goes dark again. The laptop works fine with an external monitor.
2. Your laptop turns on but the screen remains dark all the time. You still can see a very dim image outline. The laptop works fine with an external monitor.
3. The laptop screen works fine for hours or even days, but sometimes it goes very dark as it mentioned above. When you restart the laptop, the screen works fine again.

Inverter failure symptoms are very similar to backlight lamp failure symptoms and in most cases you cannot tell which one is causing the problem until you replace either the inverter board or the backlight lamp. From my personal experience, inverter boards fail more often than backlight lamps. If you have to guess, replace the inverter first.

It has bad lcd inverter and Westinghouse don't sell the parts either.

From what you are describing, it sounds like the inverter board is failing. This board sits under the LCD screen and is responsible for taking the DC voltage from the battery and AC adapter and converting that voltage to high voltage AC power for the backlight lamp that sits inside the lcd screen. Without that high voltage the screen goes dark. Most inverters do not fail right away, there are voltage fall offs etc. Inverters are approx. 15-20 bucks on ebay, for most laptops, they can be changed out simply by removing the bezel around the screen, unplugging the two connectors on the inverter and changing it out. In most cases except Dell, the inverter is just sitting in there and is not held in by screws.

your backlight is out and one of two things can cause this . 1 being the invert board is failing, or you have a couple of bad caps in the power supply. you can see the bad one they look like little cans , the bad one will have puffy tops on them ...if you see them change them or have it done if not them you will need the inverter board serviced or replaced

check your LCD inverter maybe there is loss connection or some parts damage from inverter.

If you replaced the inverter board and its still dark you need and new backlight itself http://cgi.ebay.com/LCD-Backlight-Lamp-for-Dell-Inspiron-9100-15-4-WXGA_W0QQitemZ150217509448QQcmdZViewItem

your issue with your laptop is a faulty ccfl BackLight Inverter, the display uses the backlight to illuminate the lcd screen, the inverter is used to illuminate the cold cathode light in the back of ther lcd theirs countless places to laern about it on google.
They are cheap to replace google your model with inverter at the end..

They are not too hard to replace

This sounds like bad inverter board.
Every time LCD goes from Standby to ON state - Inverter Board resets it’s self for approx. ~1 sec, during which time you should see blue screen.
Problem is that board detects problem with voltage/current on one of the seven transformers and shuts down automatically...
As to the costs:
- BestBuy repair service approx.. $600.00
- Buy new Inverter Board & repair LCD your self approx. $200.00
- Another option is to repair Inverter Board, but parts are sometimes hard to get...
Now for the DON"T TRY THIS AT HOME or AT YOUR OWN RISK solution:
-The cheapest method is to bypass protection circuit, by transferring good signal from working transformer (using 10nF/250V capacitor) to detection circuit at the faulty transformer.
Then watch movies on your LCD unlit BAD transformer totally BURNS-OUT.
This will be obvious by dark stripe about 1"-2" tall across your display.
Hopefully the only damage will be to the transformer and not CCFL (light behind your LCD), but when you are faced with a choice of using it for a while longer or throwing your LCD to garbage right now... to me this choice was simple.
Good LUCK.