Chosen Solution
The kindle is probably in sleep mode, which uses energy. Some devices do not hold their charge while in sleep mode very well. Have you tried shutting it down when you are done using it?
the dx should keep its charge 4 - 6 weeks. if it drains in 2 or 3 days with a new battery then the display was most probably replaced with an oem version from Ebay. You can open it up and search for traces like broken lids on flatcable clamps etc. Often it shows somewhere. Manufacturers can sell rejected production units online.
