1992 Toyota Celica the battery?

Electrical problem
1992 Toyota Celica 4 cyl Manual 165000 miles

Hi,
We had snow (a rarity here) the other day, and I didn't use my car for a couple of days. I tried to start my car yesterday morning and got nothing but a series of weak clicking sounds. About 2 weeks ago, I had left my lights on all night and killed the battery. I jumped it off, let it run for about 20 minutes, and it's been fine since then. So I figured maybe that weakened the battery and after sitting in the cold for 2 days, that weak batter just wouldn't cut it, so I went out and bought a new battery.A nice one with 850 cold craking amps or whatever. My friend put the new one in, and the car still won't start, even worse, the console lights (door ajar, etc) either won't come on, or they come on and then turn off as soon as I try to crank the car. Not even any clicking sounds. He tested the new battery with his jumper box and said it wasn't dead, so it can't be a defective new battery. He put the old one back in, and I got the clicking sounds again. He even cleaned off the terminal things that connect to the posts. One of them is cracked, but he said as long as they're connected, the car should start. I don't know what to do. It seems like a battery problem, but a new battery didn't solve it! Any ideas?
PS: Yes, we did try to start it straight from the jumper box, but had no luck.