How could House M.D. continue from here?

It's really hard to say. The finale was soundly hated by both fans and critics - the writers and producers have been interviewed and seemed surprised by the level of hatred. They apparently thought it was typical House fun and games. Also, Lisa Edelstein (Cuddy) quit after the finale was shot, so it looks like any plans they had as far as House and Cuddy will have to be scrapped. It sounds like a huge failure of judgement on the part of the writers, producer and the network. They'll have some time this summer to try and salvage this mess, but I frankly have no real faith in their ability to do so.