It seems you are going to more trouble than its worth using a BJT. Why not try something like this instead: -

You may need to make the zener voltage 6.8 volts to get the same activation point and, it will be equally "sloppy" in fading-on rather than having a clear-cut on/off transition but, that's to be expected with imprecise components like Zeners and BJTs.

How would I achieve a sharper switching, like the led is fully lit at 9.2V and fully off at 9.1V?

If you want a better circuit you should use a comparator and voltage reference. The gain of the transistor has nothing to do with how your current circuit differentiated between on and off.

It seems you are going to more trouble than its worth using a BJT. Why not try something like this instead: -

You may need to make the zener voltage 6.8 volts to get the same activation point and, it will be equally "sloppy" in fading-on rather than having a clear-cut on/off transition but, that's to be expected with imprecise components like Zeners and BJTs.

How would I achieve a sharper switching, like the led is fully lit at 9.2V and fully off at 9.1V?

If you want a better circuit you should use a comparator and voltage reference. The gain of the transistor has nothing to do with how your current circuit differentiated between on and off. Maybe use a comparator like the LM311: -

Image from here.

If you replace the NTC thermistor with a precision shunt voltage reference (many to choose from) and set the pot to the "trigger-point" of 9.15 volts then it should be very sharp in response. You can even use multiple comparators to get a "several" level deep LED response: -

Image from here.

It seems you are going to more trouble than its worth using a BJT. Why not try something like this instead: -

You may need to make the zener voltage more like 10 volts to get the same activation point and, it will be equally "sloppy" in fading on rather than having a clear-cut on/off transition but, that's to be expected with imprecise components like Zeners and BJTs.

How would I achieve a sharper switching, like the led is fully lit at 9.2V and fully off at 9.1V?

If you want a better circuit you should use a comparator and voltage reference. The gain of the transistor has nothing to do with how your current circuit differentiated between on and off.

It seems you are going to more trouble than its worth using a BJT. Why not try something like this instead: -

You may need to make the zener voltage 6.8 volts to get the same activation point and, it will be equally "sloppy" in fading-on rather than having a clear-cut on/off transition but, that's to be expected with imprecise components like Zeners and BJTs.

How would I achieve a sharper switching, like the led is fully lit at 9.2V and fully off at 9.1V?

If you want a better circuit you should use a comparator and voltage reference. The gain of the transistor has nothing to do with how your current circuit differentiated between on and off.