I'm having similar problems, except I can't get it to power on in any mode, whether I have the battery in or out, connected to PC or to mains. This is actually my 3rd Streak, #1 had a component burn out (which I'll get to in a second), and #2 kept overheating and restarting/turning itself off.
I think #3 has died because I (foolishly) connected it to the car charger that may have been responsible for the death of #1 (although in my defence I did check out the car charger and couldn't find anything wrong with it). It wouldn't charge from the car charger, however when I disconnected it to charge it when I got home, it did still keep coming up with the empty battery screen. When I got home and put it on charge it flashed up the soft keys for a second, then they went out. Now I'm not getting anything from it at all.
I have a feeling the phone is dead, but if anyone has any suggestions, please advise. Some more things I've checked: battery voltage is 0, voltage on battery terminals when I've got the phone plugged in is 2.8mV (sounds like the charging IC is blown), currently searching for a similar battery to try jump starting it (but with it not powering from USB I'm not hopeful), and I've completely tore down the USB cable to make sure it's all working end-to-end.