gumboot
I have a very similar setup to you. In fact, mine is worse as I have no ethernet cable at all at the TV/sky box end. What works for me is a "wifi extender". These are designed for houses where the wifi signal doesn't extend from one end of the house to the other, e.g. because it's a big house or has thick walls, or because you want internet in the garden, etc. You can put the wifi extender at the back of the TV and plug the TV and sky box into it. It works just like a router, except instead of getting its internet connection from a wall it gets it from wifi. As far as your Sky box and TV is concerned, the extender IS a router.
I bought one ages ago and my model is no longer available, but something like this should do the trick ...
Amazon.co.uk User Recommendation