So I'm assuming wall construction/thickness is your main issue then. 5Ghz will typically struggle more than 2.4Ghz at penetrating walls. Concrete and brick are the worst materials (concrete with higher water content is a nightmare).
So if you got a mesh system you'd have one connected via ethernet to your router (which then has it's wireless turned off, and typically set to "modem only" mode otherwise you end up double-NAT'ing), and then the other unit about half way between the router and the dead spot so that it picks up good signal from the base unit and broadcasts it onward.
One thing to note with mesh units is that some bandwidth is reserved so each hop will be slower.
I've been looking at the TP Link Deco over the Google Mesh system, it performs better plus you get 3 base units for the same price. Orbi is supposed to be a good performer but pricey also although personally I don't want Netgear anywhere on my network.
You could also look at powerline links and/or with wifi AP built in and just mirror your existing wireless details. But transfer rates will vary depending on your internal cabling, you never get anywhere near the stated bandwidth.