97 Ford Escort Ignition: Click, Fail, Click, Fail, Click, Start?

I have had a 1997 Ford Escort for a number of years. I’ve had a minor problem for the past two years that is now getting to be annoying:

Every once in a while–completely inconsistently–the car won’t start straight away.

I turn the key, it clicks, everything has power. I turn again, click. About 4 turns later, the locks go down and up, the alarm chirps, and then it will start.

Recently this has taken as many as 15 turns!

Battery is new, starter is relatively new, battery wires look OK at a glance but I haven’t investigated those fully.

For the starter, how new is ‘relatively new’? Is the click from the starter, or from relays under the dash? I’d pull the battery connections at the battery first, and make sure they are clean. Then, try the battery connection at the starter. MAke sure it is tight and clean.

My first swag is the ignition switch (the part with the wires, not the lock cylinder part). Here it is http://www.autozone.com/N,16400189/shopping/partTypeResultSet.htm At a price of $180, you should have someone to check your car out.
This link will show your electrician/mechanic the wiring diagram for the start circuit http://www.autozone.com/shopping/repairGuide.htm?pageId=0900c1528008debf Click on Fig. 10.