@Maxaduke 

the main thing with an extender is you need to place it where there it gets a good strong signal from the router. Place it closer to the router, sort of half way or less. 
 

Also try moving the router a bit. I found my router does better if I turn it about 45 degrees as I have a brick fireplace near by. Try higher and left and right a bit. 

  @Maxaduke 

First, I would look into the possibility of installing an ethernet cable (either interior or exterior) to your shop.  Another option, depending on your internal power grid, would be to use PLC devices between the home and shop.  The PLC device in the shop could connect to an access point if it does not incorporate Wi-Fi.  I use devolo devices here (which are more expensive than TP-Link devices) but the throughput is much greater and video streaming is no problem.

  @jra11500 Running a cable is not really an option.  

PLC is an Interesting concept.  I was under the impression that PLC devices had to be plugged into sockets that were on the same circuit.  But reading the devolo information it says it can be plugged into any socket.  Our shop however is on a completely different powerline.  It is not connected to the house at all.  How would a PLC device work in this situration, as that is where I need wi-fi. 

  @Maxaduke 

In your case, PLCs would not be a viable solution if your shop is on a different powerline.  If you decide to use range extenders, take into account that each hop halves the speed as extenders are not full duplex devices.  It's a shame that running a cable is not an option.

Just a thought...  You might be able to use 2 PLC devices (one near the router) and the other on the east end of the house.  Connect the PLC on the east end to an access point which then should cover the east end of the house and the shop.

  @jra11500 For someone who is technologically challenged, what do you mean by an "access point on the east end of the house"? 

  @Maxaduke 

An "access point" is another router (or range extender) that is configured to work in Access Point (or AP) mode instead of wireless router mode (or range extender mode).  It allows wireless devices to connect to an existing network.  If you can manage to extend your network (via cable or PLC) to the east end of your house, then an access point could provide a much stronger Wi-Fi signal in that area.