Sunchokes can stay in the ground until you are ready to use them.
All through the winter as long as the ground is thawed or anytime under deep mulch. since they don't store well after harvest, this is the best way to store them.
Jersualem artichokes will spread by the roots, with no further cultivation need.
You can feel free to harvest as many tubers as you can find. you always miss some and even marble size ones will regrow a full size plant.
A bit like a starchier potato, without all the heavy carbs.
Actually they have little starch as the store inulin, a complex polyfructose which leads to a problem in that humans can not directly digest inulin and depend of certain gut bacteria to convert it into sugars. These must be developed and cultured in the gut slowly or you will have gas pains. They make absolutely delicious french fries but slow moist cooking begins the break down into fructose and, IMO, are a bit too sweet in stews.
We have kept the same cultivar for 30 years and the only thing you really need to do is move tham to a new bed when the begin to dwindle from exhausting the soil. They don't need garden space so you can put them in an slightly enriched out of the way.
Don't forget the are a sunflower and the leaves can be eaten (even by people when young) my chickens love them. They will eat the tubers too.