"TMP Carpet" at 781-245-3939 did a good job for us, and had no trouble reaching the third floor with their long hose. The truck-mounted equipment sucks more of the water back out, so the result dries much faster than if you used one of the portable rental things.

They need to run a hose in somehow, and the process is noisy, so you may want to evacuate your cats to somewhere calmer for the day.