I am new to gardening, but I planted potatoes, also using the trench method. My plants are about 3 feet above level ground. Trenches have been filled with some mounding at base of plant. Mine also started to fall over one day, So I tied a string around them to a stake just so they don't die. I beleive you leave the flowers on the plant. Let the plant grow until it dies. I think when it flowers, it is the start of the potatoes forming. I don't plan to dig mine up until the plant is completely dead above ground. I Think mine will be getting dug up in September sometime.

There's a good series of home gardener guides at U Illinois, including one about growing potatoes.

"Harvest potatoes after the vines have died. Handle as gently as possible during harvest. Because the tubers develop 4 to 6 inches beneath the soil surface, a shovel or spading fork is a useful tool for digging potatoes.

Potatoes for use in early summer ("new potatoes") may be dug before the vines die (usually in July). When the potatoes reach 1 to 2 inches in size, you may wish to dig a few hills...

Late potatoes are usually dug in August or early September. They keep in the garage or basement for several weeks in their natural dormancy. Store over the winter in a dark room at a temperature between 38° and 40°F with high humidity. Check periodically for spoilage. Temperatures below 38°F cause internal damage to the tubers."

Here is a link that might be useful: potatoes at UI

You dig the potatoes when the vines die, as DtD said. I usually end up harvesting mine from mid-August to mid-September, depending on the variety. I always remove the flowers since it keeps the fruit from forming, allowing the plants' energy to go to the potato tubers rather than the fruit. I haven't had blight problems (knock on wood) but I try to be vigilant about potato bugs, especially when the plants are small. When the plants flop over, I usually either ignore them or use stakes and string to create a sort of fence around them to hold them up and allow me to use the path next to the potato bed.

If you want a few early potatoes you can gently root around the base of the plants while they are still growing to sneak out a few new potatoes and then replace the soil and water to settle the soil back around the roots. I haven't found this to work too well for the short-season varieties I grow with the schedule I have (too short a time between when the potatoes form and the vines die back,) but you can try to see if it works better for you.

After harvest I wash them and let them dry thoroughly in a dark place and then bag them (paper usually) and put them in the cold cellar, which stays warm enough that the starches in the potatoes don't turn to sugar. One variety I grow stores until late March, but most start sprouting in late January or February.