Does it have to be Amazon Prime? Would Sky Movies meet your requirements?
If your TV has an HDMI socket you could get a NOW TV Box. (There is actually a way to use it without an HDMI socket but things can get rather complicated!). The box costs £14.99 in Argos and that
includes your first two months access to Sky Movies. (If purchasing, take care to select the correct package as their are alternative 'Entertainment' and 'Sky Sports' passes available):
http://www.argos.co.uk/static/Product/partNumber/2757791.htm
Thereafter you pay £9.99 per month. (There's no contract. If you decide that you no longer want the service you simply stop paying). There are over 1000 movies to choose from, with 16 premières added each month.
As well as Sky Movies you'll get other 'smart' features as well (which will continue working even if you decide not to pay for Sky Movies in future). They include apps that let you watch catchup TV (BBC iPlayer, ITV Player, 4oD & Demand5) and Youtube, as well as accessing radio stations from across the world.