Question
Where do I cut the spike when all the flowers have wilted and fallen?  And how long should an orchid plant keep its flowers?

Answer
Barbara, I will assume that you are referring to phalaenopsis orchids.  Each flower normally lasts 2-4 weeks depending upon many factors.  Some phalaenopsis hybrids lose there flower spikes and they dry up.  On other hybrids, the spike stays green and may reflower.  You will notice that there are joints called nodes on the flower spike.  Since any new growth on an old flower spike will originate at a node, you may cut the spike just above any of its nodes.