How to use LM56 to switch on/off a 2kW heater
Yes, you could do that. I suggest you leave all the present thermostat and safety devices in place and only interrupt the power to the heater (just turn the thermostat to a somewhat higher temperature than you would like to maintain. 

You could drive a mechanical relay with the output of the LM36, preferably using a transistor such as a 2N4401 or MOSFET to drive the coil. The usual circuit is something like this (good for a T90 type 12V relay). 







simulate this circuit &ndash; Schematic created using CircuitLab

The idea is to avoid switching the relay directly with the LM56 to keep the power dissipation low since the LM56 is actually your temperature sensor. You could also simplify this greatly by switching an SSR directly with the LM56, but the SSR would require a large heat sink for 10A. You could use a 12V regulated wall wart to power this. 

Take all necessary care with the mains connections and have someone knowledgeable have a look at it before applying power if you're not very familiar working with mains electrics. 

A typical 30A relay will last a few hundred thousand operations at 10A, so you might want to calculate the life and decide how much hysteresis you can tolerate in the room temperature. Something like 1-1.5°C is probably okay.