The key issue to this situation is completely blocking the warm coolant flow from the cabin heater core. Assuming the blend door, its drive motor and the control switches of the electrical system are functioning properly. If any of those devices aren't work to design temperature control and A/C efficiency is greatly reduced.
The link to articles on FTE are the best approaches IMHO. They use the least amount of parts and tend to be very simply for more DIY's to accomplish.
FWIW there was a time I'd bypass a leaking heater core to avoid doing the repair and still drive the car for the time being.
This warm vent air vents (no A/C selected) is aggravating to be sure---glad to see most of us can still provide how-to-repair guides for a cure!