First off note that SPEED does not determine SIGNAL strength. You will need to determine what the highest speed router you can use is and then look for one that offers the best coverage. A crude method of coverage is determine by the antenna on a router, no antenna your looking at Omni-directional that is internal and usually good for an room or one open floor plan of a home. These are your bottom dollar routers typically $40 to 80 by Cisco/Linksys, Belkin, Netgear, etc and typically only support B and/or G protocol. If you want to make sure you have the appropriate coverage find a router that offers at least three antennas for the best total coverage plus overlap this will usually be an N protocol or AC (newest) protocol based router. Make sure your hardware supports N Wifi OR that your router choice is backwards compatible (almost all are) with B and/or G protocols. Again this is crude approximation of signal coverage of a device and not a guarantee. Another option if your technical is to repurpose old hardware into something useful like an Access Point or Wireless bridge. If you have an old Router you replaced you might be able to do this via the software already on board OR by downloading a new firmware for that hardware offering more features specifically bridging or creation of an access point.

I just upgraded my router to a netgear nighthawk x6. Not sure how many devices can be connected to it, but it's a lot. We have an 1800 sq ft house and I never drop below 90% signal. It was around $300 but worth it.

I have the NETGEAR Nighthawk AC1900 Dual Band Wi-Fi Gigabit Router (R7000). It works great but it's expensive.

Before you go off choosing an insane router what type of connection do you have and what speeds are you paying for?