How to extend internet coverage on this apt?

I've handled far worse.

And its not that simple and depends on a lot of variables. Many modern routers seem to have directional flat antennae so how you position your modem matters. You've seemingly handled the worst parts of the problem (that's an excellent position for a AP) though I'd try facing the front of the AP at the room with the worst reception. Its also worth doing your homework - I find certain brands work better than others (dlink for example, has always been an unmitigated disaster for me).

Its also worth remembering repeaters add to signal congestion - you'd want to find a less used, non overlapping channel for it. I'd try one at or just outside the room you have issues with. I just went with the cheapest one I could find, and tried plugpoints until I was happy with signal strength from both the router and the systems that needed to connect to it.

I've also run homeplug where I just couldn't get a reliable signal out. Its moderately finicky and has its own issues.