They were damaged by the birds, but didn’t detach until they hit the water. As for the hydraulics.. the APU would still have been running and supplied power all the way in.
The APU will not produce enough hydro power to successfully steer the airplane expertly the way Sully did.
My speculation, as a former AF-trained accident investigator, is that the engines possibly retained enough integrity after sucking in geese that they could continue spinning and running the hydro pumps and generators long enough for Sully to find a suitable spot.