not sure if this will help anyone, but i had same issue. tried restore - nothing, so i powered ipad off, and let it lie on the table for 2 days. After powering on, the ipad powerbutton was working again. No idea what is causing this behavior.
most likely - the power supply is a switching power supply and often there is a problem with the big electrolytic capacitors. they lost their capacity over time or "explode" (the top of them are slighty pop-out). as the capacitors are bad - the switching power supply wont start, it just keeps trying to start but the feedback loop wont let it to start - this is probably the reason why the led is flashing. so the capacitors are most likely causing the problems. if you got soldering skills or if you know someone who have, replace all the big capacitors with the same capacity and voltage ratings and it should work. if you cannot solder, just order new PSU board.
there should be black tape on certain places of the back of LCD panel. are you sure it is on all places as on the bad LCD? because the back of LCD is metal and can short something on the PSU or logic board
if you say that on external monitor it is ok, i assume that graphic chip is okay. I would go for the LVDS cable first since it is a lot cheaper, however, taking the display apart is a tough job - but if it is a bad LCD panel you have to take it apart too to replace it (if you buy just LCD panel and not the whole display assembly)
I serviced dead backlight issue last week, more on that here: http://forums.macrumors.com/showthread.p... there are three types of voltages going in the inverter- 24V DC main voltage, backlight ON/OFF signal (I refer that as backlight-enable in the post on macrumors forum) that should be 2-5.5V DC to get the backlight, and Backglight-intensity voltage (on the inverter pcb marked as VBR) that should be between 0-3.3V DC. check these and tell us your measurements to trace down the problem.
since you can see the grey apple on startup, my bet wolud be that GPU is ok. more likely it would be PSU i think - probably the big electrolytic capacitors, that would be the first thing i would go for. If you have soldering skills or someone who has, Try it as the first, relatively cheap fix - note that the electrolytic capacitors must be same capacity and same voltage and low-ESR types