The LCD display is bad, you will either need to buy a replacement LCD and Digitizer or just an LCD. Personally I recommend the LCD and Digitizer simple replacement and no heatgun needed :)

the backlight of LCD is damaged while display function is ok...so you have to replace a LCD

sherribalmer , the first thing that I would suggest is that you follow this guide. It was written for the 3G. Once you got it accomplished, re-evaluate the damage and do replace the battery. It will also prevent corrosion down the line. If you do not clean your phone now, you may end up with damage in a couple of weeks. Anyhow, once you did that see what happens. Since your screen remains black, my gut feeling would be that it is short circuited. You will most likely have to replace it. Not a terrible job to do and not really to expensive. Here is the how-to and the part is available at available at multiple places. I prefer the whole assembly since it causes a whole lot less headache and is far more convenient. If a new LCD should not work, you must remember that your phone also uses a backlight coil and backlight IC that may have shortened out. The coil can be replaced, but does require soldering on the logic board, whereas the IC requires a reballing. Hope this helps, good luck.