[tomsem]

It appears my iPod Touch (4G) is not compatible with v4.0. Oddly, it was by browsing the list of App Store updates on it that I discovered v4.0 existed (it was listed there initially), but after updating everything Kindle is no longer in the Updates list and the app remains at v3.9.3. The Description in iTunes says v4.0 is compatible with anything that runs iOS6, including iPod Touch. So maybe it just needs to clear some cache and try again. Or maybe Apple's servers are a little overloaded. I might try calling Amazon CS to see what they have to say about it.

Unfortunately, after nearly 4 years it seems the thing is nearing the end of its lifespan. The battery seems to have lost significant storage capacity, apps tend to crash more than they used to. I'm not inclined to get a new iPod Touch, but that leaves me only with the option to get a smartphone of some sort.