2005 century won't start when key turned on

Car Parked in garage. Turn on ignition switch key to start & nothing happens. Dash
lights all go on. Put lever in neutral or drive, same result. Called mechanic who came &
jumpered something on the engine & it started when key turned. Battery tested low, so
replaced it. A day later the same thing happened when wife had car at local church.
She waited awhile after calling me & went out, wiggled the steering wheel abit & it
started.
A day later same thing happened in my garage. Called mechanic. He got it started by
jumping something again. Replaced wire harness & switch to ignition key switch. Car
worked fine for 2 days & wouldn't start again with car in garage. Ignition key switch was
replaced a few years ago, when a similar problem occured. Got any suggestions I can
relay to my mechanic when I call him again? Car has 73,000 miles on it. Runs good
otherwise.