Many times a weak signal from the main router will still be faster than a strong signal from a repeater. It all depends on how much interference you have.
The best option is to use a ethernet cable with a AP in the remote room. If you do not have ethernet using powerline and hooking that to a AP or buying the powerline with a radio in the remote end is the next best. You can also consider moca if you have coax tv cables.
If none of those is a option and you get no coverage in some areas you can consider the mesh/repeater systems. It takes very careful placement of the repeater unit so it can get very strong signal from the main router yet still provide signal to the remote rooms.
I have never really looked at 802.11ax...