a pound, wow! You will be well-stocked!
I don’t have a tea ball, but from what you describe it sounds like it will work fine. I have one of those tea things that set into the top of the mug rim, and put the loose tea in its center area which dips down into the water area.
I heat the water up in the microwave until it is boiling, then take it out and the tea thing in it so that the mullein is submerged in the water, about 7-9 min. Then I strain it into another cup.
A few months back when my husband had his seasonal bronchial issues, I made a whole pot of boiling water and then took it off the burner and put in a bunch of mullein, so he had is for a couple of days or so. I’m not sure if is as potent this way though, and prefer to make it “fresh” each time.
Cheers : )
A French press will work. The tea stores well.

It comes apart in the middle to fill and then snaps back together. You hang it on the edge of your cup. Handy little gadget if you drink herbal tea that doesn't come in its own bag.

Another herb that works very well for coughs, used it with great success and it is a main ingredient in my home made cough syrup, is coltsfoot. Latin name: Tussilago farfara
Usually medicinal herbs need to be used in larger quantity than beverage herbs or black/green tea, and steeped longer. I made very strong coltsfoot when I had a cough that would just NOT let me go. I made a big pot of ultra strong coltsfoot, drank several cups one day, and next day almost no cough at all.