iPhone 4 won't power on, already replaced: dock connector, battery.

No spill damage.

I had trouble a few days ago, it didn't turn on after depleted battery, and wouldn't charge. I replaced the battery and it turned on and it was charging normally. It was an old battery that didn't last more than half a day, so it died overnight as well, wouldn't charge next morning (same story twice basically). I bought another battery, didn't work, I replaced the dock connector, didn't work as well.

I'm running out of ideas. I have a decently working spare logic board (water damaged, has occasional reception issues) but I'd really like to fix it without replacing the main board.

Does anybody have any ideas what else to try?

Also, it doesn't show up in iTunes, DFU mode doesn't work, so basically it doesn't work at all.