Place the LM 35 in the cage, in such a way that heat doesnt escape..i.e seal the hole up for the leads to exit the cage...( the sensor could be encapsulated into a 'probe that is inserted into the cage , as you would a temperature probe)
As for the electronics, use the op amp as suggested by Eric to make a comparator, this drives the solid state relay that switches the heater on/off...All of the electronics is external except for the sensor or probe as you put it.................