Good news - I've successfully tested a new web host with the kiosk system and it seems to be working fine. I have not had time to do extensive testing, but the system is on the way to working again. It will take a day or two before the LL servers "catch up" with where the new website is, but they will just start working again... soon.
I decided that waiting on LL to fix this problem was simply too risky. It's been a week already, that's one week too long for the whole system to be down. The solution basically involved a ton of work to evaluate, find, and configure a new web hosting site, which means, it's a new IP address that, luckily, the LL servers don't have a problem communicating with.
Does this mean the same problem won't happen again? I have no idea, since we don't know the cause. Once LL figures this out, I suppose they will fix the problem so it doesn't bite anyone else, or me, again.
It's been a long night, I'm tired. This whole ordeal has fried my brain completely. But I think at this point the light at the end of the tunnel is rapidly approaching... it's just a matter of time before things start working again.