Mindstorm Nxt Brick not starting
I have the same problem with mine.
Some time ago I had a really bad battery leak, and with some perseverance I managed to clean up the majority of the corrosion using distilled water, white vinegar and a mountain of cotton buds!
Unfortunately, I lost my speaker (The plastic, and thin cables completely destroyed)and I suspect some damage to the contacts along the edge of my LCD occurred.
What I find now, is that my LCD in general doesn't work, and IF my unit has been switched off for an extended length of time, I have to get a fresh set of AA batteries (All 6) pop them in, and plug the USB cable in, then just leave it sitting there connected to the PC for about 4 hours, after which I open my computers device manager, press the power key and usually see the device manager spring to life.
If the batteries are good, I might get some flickers of graphic on the display, but for the most part the display is effectively useless, however if your happy to open your unit up and not afraid of electronics, then some isotropic alcohol and a non fibrous micro cloth (Such as a glasses lens cloth) can be used to clean the rubber contacts around the edge of the LCD (Mine had 0 output until I did this, and only then did I start getting some flickers of activity)
It does seem to be a common problem too, judging by the amount of units for sale on eBay with non working displays for spares or repair...