How to watch Amazon Instant Video/Amazon Prime Video on your tablet.
Install Flashfox. I used the Pro version because I was so pleased this worked but you don't need to, the free one works fine. Tick the "request desktop site" box then visit the Amazon website and browse to your videos - voila, you can play your videos on your tablet.
How to watch Amazon Instant Video/Amazon Prime Video on your Chromecast from your phone or tablet.
Somewhat to my surprise, there is a very simple way to do this - I've been googling it frequently but somehow this has never come up so I figure I'll put it out there. If you go in to the Chromecast app on your device, there is an option to cast screen. On my phone it warns you it's not designed for this screensize but just ignore this. On my phone I can then watch via the Instant Video app or Flashfox and the picture and sound are chromecast to my TV with very good quality. I noticed a couple of stutters which I think was from the Amazon app rather than the chromecasting.
So, having finally managed to get to watch my programmes the way I want again, the next season of Lewis is apparently not available on Prime or elsewhere. I don't really want to buy the DVDs so I guess I'll just bittorrent it and wait for an electronic version to become available. Sigh. It really shouldn't be this hard to pay for what I want to watch. Still, definite progress