Replaced screen; screen turns off after a few seconds

My wife dropped her iPhone 4 (AT&T) and cracked the screen. I replaced the screen by following the directions here on iFixit. (The screen was from an eBay selling with a ton of good ratings.) The phone turns on, can be seen by iTunes, can be backed up, and photos can be down loaded--but the screen randomly turns off after a few seconds. The phone is basically unusable.

Here's what I've done so far:

1. After first replacing the screen the phone wouldn't turn on. It would vibrate though.

2. I disassembled the phone again, and found that the digitizer was not plugged in properly. I fixed it and reassembled it. (I did notice a small almost square piece of black rubber/plastic was left in the tray after reassembly. I'm not sure where it's from but it's not the rectangle bumper that protects the digitizer and LCD cables.

3. The phone turned on, but the screen doesn't stay on. After a few seconds it turns off. It's not consistent though. Sometimes the home button doesn't turn the screen on; sometimes it does and the screen turns on and stays on for quite awhile. The behavior doesn't seem to change whether it's plugged into the wall, a computer, or not at all. Again iTunes does see the phone when the screen is black. Also my wife claims that she can sometimes see the home screen icons, but very dim.

4. I tried to do a hard reboot thinking the issue might be battery related. But I'm not sure if it worked right. The screen goes black when the Apple logo appears. Then it was hard to get the home button to turn the screen on again.

5. I tried to do a back up and restore just in case it was a software issue. No change.

So that's where I'm at. Any ideas? Could the digitizer still be improperly plugged in? Could that little piece of plastic be important? Did I mess up the motherboard on accident? Could it be a battery or proximity sensor issue?

Help!

UPDATE: Something did change after the back up and restore. It almost works... When I set the phone or tap it with a little force, it turns off. Maybe somethings loose?

UPDATE 2: First off, what I've been calling the digitizer connector is actually the LCD connector. I had the two confused. The LCD connector is what I'm having trouble with. It just doesn't click in very well. Could that be my source of my problems?