I bought my iMac 2 years ago, it came with a Magic Mouse, within several months I started having the same battery issues. I looked here, and on the Internet to find out if the problem was already known about, but found only people guessing wrongly at what the problem is. I do hope Apple has fixed this, but unfortunately many of got stuck with a junk Magic Mouse. As an Engineer with 25+ years of experience in the design of electrical contacts & connectors which includes over 2 dozen patents ... believe me when I say the problem here is that the recessed positive contacts (at the on/off switch end) are too deeply recessed to make contact with the industry allowed variation in AA batteries. Those contacts are too deeply recessed because the plastic is too thick (in the hole that forms the recess) and the metal contacts are not tightly up against the backside of that plastic wall, in fact it looks like they are installed with a gap and/or can deform/bend away further opening that gap even larger. As mentioned above many industry AA battery standards define the positive button contact height as being 1.0mm or larger ... however that plastic wall is about 1mm thick, add the gap for the metal contacts and you have no electrical contact no matter how much contact spring force there is at the negative end. Unfortunately, there is no simple good fix for the average person to perform, the recessed contacts can't be accessed to bend them to remove that gap and the pastic is too thick anyway .. leaving us with a "duct tape" fix, like wadding some tin foil into that positive terminal recess and swearing when it falls out during battery changes. A nice elegent fix, would involve disassembling the mouse, machining plastic, backing up those terminals... absurd right?... would be easier if Apple sent us a new mouse that works.
This problem has nothing to do with overall battery length, battery diameter, or inadequate spring contact force from the negative mouse contacts. Also I doubt the brand or type of battery is the issue for anyone, the positive button heights vary among brands/types, but probably are 1.0mm or larger per the industry specs, Apple just didn't design this mouse to that industry spec, or they did not control their manufacturing quality to yield what they designed. This iMac was my FIRST Apple purchase after decades of avoiding and dismissing Apple, ironically they won me over with their progress and high quality products ... I'm rather disappointed.