"High speed broadband" (anything much above 4-5Mb) usually won't take telephone extension cables - at best you get severly reduced performance but usually they just won't synch at all. If you have wifi range problems at the opposite end of the house then either run an ethernet cable to a second Wireless Access Point (WAP) or use a decent Powerline system to run the signal to other WAPs. "WAPs" are just wifi routers/switches and can be either specific products or tweaked routers. WHen we upgraded to fibre internet I reconfigured the original router to be a WAP (you can find pages of advice on how to do this on the interwebs), but as my home is two 200year-old cottages knocked together the walls are mostly flint which is wifi-opaque. So I've just put poweline WAPs in almost every room (each with a different SSID) and devices just "roam" between them.
PDR