By starting the APU and bringing the APU generator on line, the crew of 1549, kept the normal flight control laws (more importantly, the flight envelope protections) available.
This was important, as without auto throttle available (engines flameout) and the neutral speed stability of the C* pitch law, it is not possible to trim to an airspeed. Sully was attempting to maintain green dot (approx max L/D airspeed), but because of the workload and distractions, got slow.
As a result, the last ~200 feet of the flight was on the AOA limiter, which prevented a stall/departure.
However, because he was on the AOA limiter, there wasn't enough energy to flare, which resulted in a hard touchdown and damage to the lower fuselage.
For details, see the NTSB report NTSB/AAR-10/03, specifically the analysis part starting with Section 2.3, Flight Crew Performance.