What to do when an inspector misses stuff?

So we just bought a home, and upon moving in we found that a tile in the ceiling had fallen, and that the wood above it had been rotting.

Now we closed on the house 2 weeks ago, and the inspection took place less than 3 weeks ago. My wife noticed the roof looking discolored and caving in a bit, but the inspector said that it wasn't a big enough issue.

Fast forward to yesterday, we had the AC guy come in and check the pipes (because we thought that the leaking was coming from the A/C), and he pointed out that the leaking was coming from the roof, and not the A/C. I called the home warranty folks today, and the roof isn't covered. I called the realtor for advice, but she isn't answering. I hate to think I am on the hook now having to buy a new roof upon buying this house, when, had I known that the roof was bad I wouldn't have purchased the home in the first place.

The inspector messed up. I am not trained to spot these things but he is, and he dropped the ball. What can I do here?