Hi Luke,
as I have a really short battery life for my iliad I have read this thread with delight as it sheds some light on my problem - I have a Iliad Book Edition and according to your step by step How To I have measured the voltage of both batteries, which are 3.99 and 4.18 respectively - this is different, also the Battery with 4.18 V is somewhat inflated, you can push around a bubble of gas in the envelop.
So I would assume that my batteries are also gone. Do you agree?
You write that the batteries cannot be changed, but there are numerous postings in this Thread
https://www.mobileread.com/forums/showthread.php?t=30794 regarding the change and nobody seems to put an emphasis on the V2 and V1 difference as you do here. Please verify that a battery change is not possible for V2 iliad readers.
Best Regards,
FG