Can someone offer suggestions on how to repair this issue?

sorry your freezing so ill give it a shot --here we go --- pop the hood and go to the pass. side and up agianst the firewall is two pipes those go to your heater core where they come out they connect to a hose there will be two and are abt. 3/4 or so in diamiter-- the heater is off so look at what position there bolthin this is a valve that is the on n off ttheres a cable w/ a wire on bolth what you want to do is manauly by hand move the valaes bolth of them open . you following me here you will move it 1/4 too full try that but if your heater core starts too leak on the interior floor pass side shut off motor and put it back rember move the valves too the same positions so there the same and be careful dont attempt it when motor is hot or running and check radiator water level is a must try this and be careful im not responsible if you get hurt so good luck and warm up

"Best" involves replacing parts. Dealer's expensive; junkyard involves your own work -- you pick.

"Easiest" might be as simple as shoving the broken lever with some other implement, like a screwdriver -- but it's "inelegant!"

broken on the heater box, or on the dash control?

call a junk yard or auto salvage yard and see if they one