The less heat loss from the furnace body itself will increase the efficiency of the furnace. Now as to wrapping it will insulation to cut down on the heat escaping would be okay as long as that is the problem. With any hand fired furnace, you have a large door that radiates heat into the room, you can not get away from that. If the majority of the heat gain is coming from the door than you are not going to gain much by insulating the body.
If the shell is warmer than the air temperature than adding a layer of insulation to the body would cut down on the heat loss into the room and since heat rises would in turn cut the heat infiltration into your daughters room.
I would insulate with fiberglass board insulation not roll, it can be made to fit tighter to the furnaces shell than the blanket type does.http://www.buyinsulationproductstore.com/servlet/the-487/Certainteed-Rigid-Fiberglass-Board/Detail
And remember any exposed uninsulated duct work and add to the heat in the room from air leaks and radiant energy.
What it all comes down to is how much heat is lost into the room around the furnace.