i have a 2000 integra as well. i was in an accident and my horn went out. I've already replaced the right fender but my horn still doesn't work. I just haven't taken the time to look at it, but i'm hoping i just blew out a fuse or knocked a wire loose.
That clicking noise you're hearing inside the car is the relay in your dash. so that's a good clicking.
Tomorrow morning i will take a look at my horn and see if i can fix it and i'll get back to you.
I agree with tegbooster that this would be a simple project for you, a great starter project to help get you under that hood. Your horn is visible from the outside of your car on the passenger side. just look up from the ground and through your front grill and you should be able to find a round metal thing with a whole in the middle of it.