Looking for a single router in this case, would be rather difficult. Even if you did mount a high powered router about mid-line to the main level of the house, WITHIN the house, you would usually only cover about 400 feet (not square feet), INDOORS. This is because how the signal bounces around, and hits, goes through walls, etc.
For those of us not using the metric system, the house in question is about 5300 square feet in size. Normally, a networking solution would require an externally placed router, either on a support structure (another building), or a pole outside of the house.
For a multilevel house, that is about 5300 square feet, I would recommend an externally placed Cisco type router (E4200 or newer), that is mounted about 75 to 80 feet parallel to the mid-line of the main housing unit.
This would provide ample coverage to the entire structure, and only require the use of one router.
As to the antenna, a directed antenna (not omnidirectional) is probably best in this type of situation. Directed antenna's for externally mounted routers, will ensure that the signal is only directed at the location in question (the house), and not waste energy in directing it in all directions. The cisco routers can use high gain unidirectional antenna's without much issue.
If coverage is needed, required, without signal degredation and loss, you could also consider a second unit equidistant to the first, transecting the path of the signal. Note these units would act as Access Points, not as independent units. Both can be tied back to the central internet access WITHIN the house, and even in the case that I have done, to a third indoor router. The point is, they would all be acting as one, providing the roaming coverage you need.
Forgive the crude drawing lol:
Typical Large home setup:
Mission Critical (building, hotel, hospital, etc):
INTERNAL ONLY: Let us assume that we wish to do only an internal network, and nothing external. A single, high gain, omnidirectional antenna on a high powered router, would not cover the entire range of a 5,000 + square foot house (these plans are for a 5300 square foot home):
To do an indoor option, you would need to consider your layout; the router should not be placed on an external wall, nor should it be placed under stairs, or by a metal barrier. On each floor, the router should be equidistant from each other router in the home, meaning you should design in a diagnal, if possible, or in a zig-zag fashion; where the primary router is on the second floor, and secondary and third routers are on the first and top floor as diagnals from the first:
However, the best arrangement for a large house, is still the external CPE:
The bar there, is the signal strength.