I would turn the key and hear one click and nothing happens.

My 97 Camry LE, now with 278k miles still runs great, but over the past 2 years I would occasionally have trouble starting it. I would turn the key and hear one click and nothing happens. This usually happens every 4-6 weeks or so and only for 3 or 4 days then back to normal. I never took it to my mechanic because until today it was always easy to get it started. I would only need to pop the hood, jiggle the red battery cable which would set off the alarm, then press a button on the remote to kill the alarm and it would start right up... Sometimes it takes 2 or 3 tries.

Today this happened again but I could never get it started. I turn the key to the on position and the indicators on the dash are bright, the headlights are bright and the horn blows like normal. Then I try and crank it and hear one click, then nothing. Oddly, when I release the key the red indicator light on a button next to the steering wheel starts blinking, indicator lights on the dash blink faintly and I hear a recurring buzzing/humming noise coming from the fuse box. I removed the fuse box cover and unplugged each large fuse one by one and none looked bad. The buzzing noise continues so to keep from draining the battery I remove the positive cable from the battery terminal.

Whenever I decide to try it again to see if will start, each time I reconnect the positive cable to the battery terminal the alarm sounds. I kill the alarm with the remote, get in and turn the key and it does exactly the same as I stated earlier.

If anyone has had the same problem and/or knows what the problem may be and how to solve this, I am sincerely grateful for any helpful advice/suggestions on solving this problem.

Thanks!