I am not sure that rechargeable batteries are a full solution, although probably not a horrible move.
Most rechargeable cells don't have the full capacity that premium alkaline cells, or alkaline/lithium cells.
I use an electro-luminescent remote for my home theater that chews up new-generation rechargeables in less than 2 weeks, and I have two full sets of batteries, that I was swapping in and out.
A set of alkalines that aren't shelf-life depleted usually last well more than a month, to two months. the good lithium ones, rated for electronics last even a bit longer.
With the wireless keyboard issue, I wonder if the MagicMouse is not sleeping the same way, and keeping the bluetooth transceiver more active, more of the time... and thus that keeps polling the keyboard more often, keeping the keyboard's transceiver more active more of the time, as well.
Because, I doubt that the keyboard and mouse correspond with each other. they both correspond with the computer's transceiver, which may keep them all "awake" longer.