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[Kep]

Too many options floating around that need to be eliminated rather than just guessing.

I think you can eliminate a worn ignition switch ("key") because when you turn the key you got a click. That click was the starter solenoid actuating. It also exonerates the wiring from the ignition switch to the relay.

You mentioned that you replaced the positive cable and cleaned the battery posts. OK, I am guessing that you thoroughly cleaned both sides of each connection at the terminal and also the post on the starter. If so, and if the connections are nice and tight (can't wiggle them with your fingers),then you have exonerated the positive cable. Have you cleaned the negative cable ends as well?

jtex2717's points are possible as well. A weak battery can result in the one click scenario. If you replaced it with a new battery and it worked well initially and then got worse then his comment about the alternator is well taken. When you try to jump start a completely dead battery it often won't work until it is charged up enough and that can take a long time (or maybe never if a cell or two has been damaged). Since you can't drive the car to the parts store, one more test would be to remove both cables from the suspect battery and jump from the donor car directly to the pos and neg cables. (Being very careful not to allow the positive clamp and cable to touch anything. A rag would help.) You can do significant damage to the car by disconnecting the cables before you turn the car off. Turn the car off before disconnecting the cables in this setup. This takes the battery completely out of the equation. If it does start then your battery was bad (or maybe drained) or your alternator is failing (thus allowing the battery to drain). Again, the best way to determine this is a free check at a car parts store but since you've waited until it won't start at all you may not have this option. If you have another vehicle that you could swap batteries with it would also give you some excellent information.

I can't think of a direct method to check the alternator at home. An indirect test would be if the car started with a good battery and then, after time (several hours of operation or a lot of starts), the battery drained to a point that the car wouldn't start. Then charge the battery back up and if the car starts up again, the alternator is suspect. A battery charger (at 2 amps) should charge a good battery back up over night.

Single clicks are often a sign of a bad starter relay (the contacts wear out). Use Mike Gerber's test to check this possibility. This happens a lot to our Camrys.

Good luck. You can do a lot yourself or a mechanic will figure it out for you.

Kep