How long does the cake keep, and what is the best way to store it?

Mme. Heatter makes no recommendation, and she would if there was anything special about it. Because of the glaze this won't have a problem with drying out, and there's not much else that could go off- it should be fine for at least 3-4 days, and may improve with time. Airtight storage would be best, but at least something with a well fitting cover