I've had my E4GT for a year now. The battery life is pretty bad and I'm still on stock Gingerbread. I've never upgraded to ICS so I don't know if your battery problem is worse but I can never make it through the day and sometimes I'm down to 20% by noon.
I can usually get it to go to deep sleep just by toggling off the screen (tap the power button). CPU Spy verifies this. Just reset the timers in CPU Spy, go back to home and turn off the screen by tapping the power button, wait a bit, then go back to CPU SPY. You'll see if you achieved deep sleep and for how long.
If you think it's an app causing the battery drainage, use GSam Battery Monitor to see which apps are using the most battery. Both CPU Spy and GSam are available on Google Play.