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I can get a power supply in there. This might sound very stupid but would have extension lead work as a short term solution until we get an armoured cable fed down there?
Getting a Cat5/6 cable IS possible but quite an effort as I'd need to route it up into the loft from our living room (location fo the router) back down and into the garden through some trucking or some such...
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If you are getting a proper electrical supply in then using Powerline solves the problem.
Are you just using the 'shed' for normal browsing? No intensive data?.
I wouldn't recommend a extension cable 'out of the window' (certainty not without a RCD) but I'm a sparks. And Powerline systems don't like extension cables.
Have you got a power supply in there? If so then use a Powerline system.
Can you feasibility run a CAT5/6 cable into it?
If none of the above then WiFi repeaters as close as you can to it indoors.