potato harvesting

Hi there, i successfully grew a whole patch of potatoes and have been digging them up as needed for the last 2 months or so. As soon as the plants completely died i cut them back as the plants grew to over a metre tall and there was straggly brown stems everywhere! I still have 5 plants in the ground and with all the rain we are having in Wales at the moment im wondering whether its best to leave them in the ground and continue to dig up as needed or dig them all out now and store?

Any advice?

Thanks!

Zoe