Is strange, because the laptop battery should do the dirty job, the car just recharges it. So even with dirty AC input, the thing should at least turn up! (laptop battery remains as the main source)
Check if the laptop battery is working!
Check the polarity of your car gadget! Check the attachment to see what I mean
Turn on your laptop using home AC, under Power management Menu select the "Economy" mode, and turn the LCD backlight brightness to the minimum
So, the load over the laptop battery will be easier, so recharging it.