Ok, reporting back again. I reported earlier in this thread that I've had the same problem as everyone else. Now, exactly two weeks ago I simply switched back to using my old Logitech mouse instead of the Magic Mouse and put in fresh batteries in my keyboard at that time. Before, when using the Magic Mouse, my keyboard batteries would have definitely been dead by now (usually only lasted a week). But, now using my Logitech mouse for two weeks, the keyboard is reporting a 77% charge remaining.
So, I say this pretty hard evidence that the issue is with the Magic Mouse.
I think I may now try installing the mouse driver separately and see if that helps as described by the poster a bit earlier in this thread.