Excellent item and gives great coverage. ideal for a house with children as i didnt want the ethernet cable laying around to be tripped over. Now for the techy stuff...
This wireless adaptor will work with any wireless router as it is compatible with A B G and N routers. As to whether you are able to take advantage of Wireless N Technology is dependant on your router. the Thompson Speedtouch 585 is a G router, so this wireless adaptor will work just fine, just won't benefit from the 'N' capability. This is the case for most routers supplied by your service provider. BT's homehub 2.0 or later has 'N' technolody and so it has the benefits or greater distance and connection speeds. still, 300Mbps is way faster than you need, the only time you would use this would be streaming uncompressed HD video through your network to your xbox. the main advantage is the greater coverage, if you are experiencing signal drop outs due to your router being far away, if your router is 'N' compatable then this adaptor should fix your problems.
as for speeds to watch SKY PLAYER on your xbox, your video stream is clamped by your broadband speed. a 'G' connection offers up to 54Mbps where as your download speeds will normally never be above 20Mbps and on average it is around 5Mbps (unless you are lucky enough to have the top virgin package and even then their max is 50Mbps). In a nutshell, SKY PLAYER will work on either 'G' or 'N' connections, it is your broadband that needs to be fast enough. i run 1 Meg from BT and it's not good enough.
i hope this has helped answer your questions and for those looking to buy this product, it deserves 5* all round apart from the price.