The biggest problem you'll have is that the LED Cinema Display has the Mini Display port connection. Unless you can find a converter to connect to the VGA port on your laptop then give you a Mini Display Port socket to connect the display to then you're out of luck.
In addition to this, VGA connections don't support the type of resolution that the Apple Cinema Display can show. It's likely that your laptop won't be able to display a suitable picture on a screen of that size, or if it does it'll look terrible.
There is no simple adapter that will convert an analog VGA signal to a digital signal (mini-displayport or otherwise). You need an active analog-to-digital converter and I don't believe such an animal exists for VGA because there isn't enough information passed through vga to be able to convert to a digital signal.
As mentioned, the biggest issue is that the VGA signal is analog, and the Apple Cinema Displays only accept digital signals. Does your laptop have any other videop output ports? A DisplayPort or Mini DisplayPort would be the easiest, of course, but if you have an HDMI connection, you could get something like the Atlona HDMI to Mini DisplayPort Converter and Scaler. Not exactly cheap at $170, but if you really need it, and you have the port, maybe it is a solution for you. The only issue, it seems, is you would be limited to 1920x1080 resolution.