The right hand view above is what happens to the lifting area if you vary the ideal speed – it moves towards the retreating blade side, and when it reaches the edge, you get your VNE for autos. The best lift/drag ratio in autorotation is obtained at best endurance speed, whatever that is (check the manual, but most helicopters are designed for a speed of about 45 kts).
Try to establish the cause of engine failure—if it's fire, close the throttle, but, if not, consider initially closing it only to idle speed (or not closing it at all) as the engine may be able to provide enough power to help you out a little, but this point is controversial as many people think you should secure the engine and fuel in case the landing gets hashed up, which may actually be caused by a playful engine, or a fire caused by a hot one (turbines cool down quite fast after they are turned off). Also, if the main drive shaft breaks in a 206 or a 407 (and others), you will need the engine to drive the tail rotor. However, the discussion below will consider it closed.

Several factors may affect your rate of descent, such as gross weight, air density, airspeed and rotor RPM. Changing airspeed, though, is about the only one you have direct control over that gives you some flexibility, as the RPM must remain in a small speed band to be effective— remember that only the portions of the blades between 25-70% of their length provide any lift. In any case, every 1% reduction in rotor RPM results in a 2% loss of thrust, which will be the same as if somebody threw that weight in the back of your machine. The other significant point about keeping your RPM up concerns the tail rotor, which runs at a fixed speed relative to the main rotors – if they go slower, the tail rotor does too, and loses some of its effectiveness.
Changes in airspeed can have dramatic effects on the rate of descent. If the recommended IAS is 60 knots (fairly common), for instance, speeds of either 30 or 100 could increase RoD to as much as 3000 fpm, because the lift vector is reduced when you alter a relatively horizontal rotor disc, so if you want to change your angle of approach, don't forget to use collective to compensate (when going for range, you must use both collective and speed to get the full effect—you can pull collective until you get nasty noises in your ear). As forward airspeed increases, the driving region of the autorotating blades moves towards the retreating blade side – the point where it meets the edge of the rotor disc is where power off Vne is found. If you go beyond it, your driving region gets smaller, so your rotor RPM will decay.