Thank you, Everyone, for your suggestions and kind words!
Just an update: Liam woke up this morning, and for the first time in 10 days or so, he's no worse than he was yesterday. As far as I'm concerned, that's AWESOME!
I think he's on the road to recovery, though if you could see his little body, you'd realize that it's going to take a LONG time.
The worst part is not knowing what's wrong and if it's something that can be prevented in the future!
BTW, the boys have spent about five hours in the pool today. If chlorine can knock this thing out, we'll have it licked in no time!