OK, I have had a look at your site from the outside and it looks very nice. You have 5 sites set up?
Have you run the updates page to update the network of blogs?
I also crawled over my version of update-core.php but cannot see where this could be failing.
As Ipstenu says some php errors would help see where this is crashing out. I suspect that there is a linking problem. Somewhere there is a link URL which is incorrect but it could just as easily be a fouled up MySQL call.
If it is a MySQL call which is failing then bringing the network up to date 'should' fix it.
One of the beauties of WordPress is that you can completely delete ALL the folders, re-install them and it will work fine. But you will need to backup the core part of the install.
This is really only .htaccess the config.php file and wp-content directory. Everything else is essentially a standard WordPress install.
Because I don't trust computers at all I would also backup the entire MySQL database before I tried this out so I could restore it in the event of a failure.
You may find that you have to back up the database in bloks rahter than all in one go, they can get pretty big.
Good luck with all that. There is a fix here, we just have to find it.