Its Not turning on OR charging! Help!

Ok so heres the deal,

I got a Like new PSP two weeks ago, my Friend who gave it to me said He's rarely used it. So much so that the Battery was puffy! I took the Puffy Sony battery out, and ordered a New Battery and a AC power cord. A week later, I get the package, and I was crazy excited to boot her up for the first time! I grab the new battery and Put it in, and grab the Charger and plug it in as well...I let it charge for 3 Hours. Whats real strange here is that the orange indicator that would be on...isn't. And the PSP was actually HOT, and the charger for some reason doesn't go in all the way. I try to start it up...and Nothing. I take the battery out, and blow on it, put it back in. This time I push it in hard, and viola! It Starts!!! Just a few seconds later, it Goes black. I take the battery out & let it sit for half an hour. I throw the battery back in. It Starts once again, but ONLY if I push it in. It lasts for 30 seconds. and cuts me off once again! I get frustrated and try to get my mind off of it. I can't. I've waited an entire month for my PSP, and to see that it doesn't work just infuriates Me! Its been a week since the incident and it still wont charge. I'm to the point where I'm just about ready to get inside there and see what the real issue is. Any Ideas?!