Most clubs that regularly offer live music have a booker -- a person whose job it is to book acts for the club. Most bookers do other things for the club, too, but they usually have regular hours during the week to take booking calls, and that's the only time they'll talk to you.
To find out the normal booking hours, look on the club's Web site or call and ask. Call back during normal booking hours and pitch the band. You'll be saying the same kinds of things you've put in your demo package cover letter. The goal of the pitch is to get the booker interested in your band -- interested enough to want to check out your demo.
Some clubs get so many demo packages, and have already booked things so far out, that they won't be accepting demos when you call. But if the booker asks to see your demo, send it out immediately so that it arrives the next day. Then, sit and wait.
It is not unusual for a booker in a popular club to get 25 or 50 demos a week. Do not call the booker the next day. Wait for a month and see if you get a call. If you haven't heard anything after a month, then you call and see if anyone played the demo. If no one listened to it, pitch the band again. Then sit and wait another month.
If you're getting the impression that making it into the club scene is pretty hard, you're getting the right impression. You have to be willing to do the work up front to have a credible band and presentation, and then you have to be ready to accept rejection. There are a lot of bands out there. Having a big following is the best way to break through that crowd.