Sorry, I think I may have misread your initial post. I was assuming that you were using the Adobe Digital Editions (ADE) software that runs on a PC or Mac, and then transferring books to the tablet. This is the Digital Editions forum, but I see somebody else moved your post to this forum. Anyway, I guess it should cover anything to do with Adobe's horrible DRM infrastructure as well as just ADE specific problems.
Can your please just confirm my reading of your second post: you are both just using tablets and downloading your library books directly to the tablets, and not using a PC or Mac at all as part of the process?
In that case, ignore most of my previous post. My reference to 'start ADE' was starting the program on Mac or PC.
I haven't got an Asus tablet, and it looks as if the "My Library" software is different enough from what I have tried that I can't give any direct advice.
Doing a quick google search came up with a thread that may be useful:
and which included a pointer to an issue a bit like yours
Good luck.