A JFET based light sensor
The battery polarity is drawn differently than it's said in the question text (noticed already by commentators). If we assume the text is what's wanted it will not work. LDR resistance variations do not affect Vgs because there's no gate current ==&gt;&gt; light variations have no effect to the fet conductivity, the current of the led stays the same in dark and in light.
You need a voltage divider to generate voltage variations with LDR. If you use a bipolar transistor instead of fet you can control the base current with light on LDR.
You can set easily predictable voltage treshold and hysteresis if you use a comparator IC to sense what voltage a voltage divider with LDR outputs. Hysteresis is needed to avoid ON-OFF oscillations near the treshold.