Might be worth checking or changing the WiFi Channel first Rob, If its going from full strength downstairs and then goes to 1 bar in one of the bedrooms, could be its being hit with a co channel from a Neighbour somewhere.
Failing that, as said 10m will be fine on cat5, 100m is the Cert spec, although Ive run 400m in a Quarry once and got away with it (had no choice).
You could just run a 10m Patch Lead (if you can get the plug through the holes, or remake it the other end) or if fitting a RJ45 Socket at the other end, just make sure you follow the termination rules about keeping the twists right up to the punch down.
The 240v relays do work but also have irritations even at short range.
Another option is to get your hands on the older type Routers that let you put them in repeat mode (WiFi Signal Repeater).
If you go for the Cable option, give me a shout Rob, got loads of near empty boxes here that I can pull some off and send to you, in case you need to go and buy it.