I have a summer hose at the back of my garden, when I had this build I had a electrician run cat5 cable from the back of my main BT phone socket to the summer house. He tested the socket in the summer hose with a phone and it worked. I have now purchased a TP-Link wireless n router and want to make it work in the summer house. I have a BT smart hub connected to the master socket in my house for internet in the house. How can I get the router woring in the summer house?
Well I guess we have at least a CAT5e cable run from the house to the summerhouse which for some obscure reason is wired just as a remote phone extension. If indeed it is CAT5e cable run, then it needs to be rewired with a CAT5e socket on each end. Then we can start to provide the service you require.
To provide an internet wireless service in the summerhouse will mean a patch cable run from the socket in the house to a spare way on the router. At the other end a dedicate wireless adaptor (AP) is used with another patch cable from the summerhouse socket to the AP. Job done.
With the appropriate skills an unloved, wireless router can be configured as just an AP service.
If you want a phone extension in the summerhouse as well, then a pair of CAT5e Voice/Data Economisers will be needed plus a BT to CAT5e patch cable and a Master Line Adaptor BT to RJ45. (These items not required if it's a VoIP phone)
I do, however, have extreme doubts that the existing cable run is CAT5 and is more probably CAT3 or similar.