In my opinion, it has nothing to do with hooking it up to the internet. Why do I think this? Because it sounds a lot like the problem I had. The problem is probably your DVD-ROM drive. I swapped out my DVD-Rom drive with a new one on the same machine and then it read it fine. My DVD-rom drive didn't have any problems reading a 360 game (well, I did get some dirty disc errors and such before this). I had to move my machine to hook up to the internet for a while, and I think the moving may have hastened the DVD-Rom problem. It is most likely your DVD-Rom if you get the white Xbox screen and then it won't proceed, or if it just won't read the xbox game at all, but will read your 360 games. You may also get the dirty disc error. Alot of 360s get bum DVD-ROM drives. And the Red ring of death. Quality control on the 360s suck. I am on my 3rd one. Good thing I can afford them now. Back in college, I would have been screwed.
I may be wrong about your situation. If you really think it is your because of your files are bad, try this: delete the NFL 2K5 file on your memory card or hard drive AND clear your hard drive cache. Do a google on how to do this, because I can't remember right now. It is something like LB RB x LB RB, at the hard drive screen. You must clear the cache to delete the update that allowed you to play NFL 2K5. THIS WILL CLEAR ALL OF YOUR PATCHES HOWEVER. So you will have to put all of your games back in after your are hooked up to the internet again to get all of your patches back. MS didn't come up with a great way to put patches on.
If it is indeed your dvd-rom, you can buy one on the internet and swap it out. There are guides on how to do this. Tread carefully here though or you can ruin your 360, like my friend did.
If you end up buying another 360 down the road, do a google on how to find the new chipset. It runs cooler and is a quieter machine. That is what my main 360 is now. It is indeed quieter (though it is still loud, but so is my PS3 when it gets hot), and it is definitely cooler. I have not had ANY (knock on wood) problems with this xbox 360 with the new chipset. Look out because Best Buys and many retailers are still trying to dump off the old chipset.
Edit: Forgot to mention, first try cleaning the DVD-ROM drive by opening the unit and using a swab and cleaner (alcohol works) on the lens. It probably won't work, but this did work once on my xbox 1 when it would not read discs well. I ended up having to swap its DVD-rom drive out as well. Opening the 360 is a real BITCH. If you are not a handy person, don't bother with any of this stuff.