This is [hopefully] another way of doing it, that gives the brightness proportional to temperature; excuse the quick & nasty sketch, I was trying to emphasise everything as I did not know how clear a pencil drawing would come out in a photo..
The thermistor is the circled resistor.
The zener provides a fixed reference voltage to compensate for battery voltage variations; a "9V" battery drops to near 6V over its lifetime.
The circuit is in principle a voltage controlled current sink; adjust the preset so there is no LED current when the thermistor is cold.
With the thermistor warm, the emitter should be at around 0.3V and the 15R emitter resistor sets the current at that to around 20mA.
You can change that to vary the "warm" brightness.
The upper resistor in the LED feed is idiot proofing, as without that the current will continue to increase with temperature - if it's eg. ever left leaning on a heater, it could fry the LED with excess current.
That extra resistor should be selected to match the maximum wanted LED current, based on a 6V supply so it does not limit as the battery voltage reduces.
There should also be a resistor from base to emitter on the transistor; I forgot that when drawing it... Use something like 100K or 1M, just to bypass any leakage currents.
You could use a darlington rather than the single NPN transistor; if so, omit one of the base diodes to keep the same voltage relationships.