Well, after messing around with my iPad 3 for about an hour, I was able to fix the problem where the button would not put the iPad to sleep or wake it. I am not if the this solution will work for anyone else or not, but I thoght I would post it anyway.
All I did was go into the options and turned off my Passcode Lock (effectively removing the passcode to unlock my iPad). I had an exchange server email account from work that required the passcode feature to work, so I delete that email account before it would let me turn off the Passcode Lock. Once both the exchange account and the Passcode lock were removed, the sleep/wake function of the power button immediately started working again. I then went back in and turned Passcode lock back on and re-added the exchange account and the sleep/wake function still works. So, it looks like the problem was a software glitch (at least in my case). Also, the Lock Screen button on the Assistive Touch menu now works perfectly fine.
Anyway, I hope this helps anyone else out there that may have problems with the sleep/wake function after replacing their iPad screens.
Thanks,
DJSin99
**** UPDATE / EDIT *****
Well, I started doing some more digging and I don't think it has anything to do with the Passcode Lock or the Exchange email. I forgot that I had turned off the "iPad Cover Lock/Unlock" option while I was trying to figure out a fix for the problem. After I got things working, I went back in to turn it back on so my case would sleep/wake my iPad when I opened or closed the cover. As soon as I turned the "iPad Cover Lock/Unlock" setting back on, the power button quick working to sleep/wake my iPad again. As soon as I turned the option back off, the power button went back to working normal, so I think it has something to do with either a glitch for that software setting after a screen is replaced or something with the mag sensor being damaged before/during the screen replacement. Anyway, good luck to others trying to fix similar problems.