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Almost all airplanes hit the runway vertically less than a 100 feet per minute, which is like less than traveling 2 feet per second. However, the horizontal speed varies greatly. Glides can land at speeds below 40 mph, while large comerical airliners will land at around 150-170 mph. It depends on the design of the wing, the wing loading, and most importantly, when it will stall. Almost all aircraft basically stall (sometimes fully, sometines partically) upon landing.
As indicated in the question, the speed of sound at 35,000 ft is 663 mph (1,067 km/h) according to the standard atmosphere model. Even if the plane's ground speed is over 700 mph, however, its airspeed will still be only 560 mph. As a result, the aircraft's Mach number turns out to be about 0.84, or 84% of the speed of sound. In other words, the Boeing 777 never flew faster than the speed of sound during your flight.