Hello and good morning. You are not being foolish in any way. that part that broke off is actually the one tube from the heater core itself. That plastic piece used to be connected to the remainer of that tube. You will need to replace the heater core itself to fix this. Before you get involved, the entire dashboard needs to be removed in order to remove the heater box to get to the heater core. Its a very large in in depth job. Is this something you feel comfortable tackling?
Your JA Expert,