Laptop powering up intermittently

I gave my daughter a laptop about 6 months ago. It's been working fine until about 2 weeks ago. She told me last night that the power won't come on. I played with it for about 2 hours getting it to come on every once in a while. While the computer was on and in Windows I checked to see if the battery was charged, it was.

I took the laptop apart and checked all the connections. I re-applied thermal grease to the cpu. Made sure the fan was working. Made sure nothing was blocking airflow, etc... I did check the RAM in another computer (before I got it to boot), it's ok. Couldn't check the hard drive in another computer, but I know it's ok because it finally booted.

What advice would anyone give? Maybe the Battery? Maybe the Power Supply? It feels to me as if it's a power problem, but I'm not sure what to check next.

The specs on the Laptop are:
DuraBook
AMD Sempron 1.8ghz
1gb PC 333 DDR 2700
80GB Fujitsi Hard Drive
DVD/RW Dual Layer
Windows Vista Premium upgraded from WinXP Home

TIA bugspop1

Is where the power plugged in seem OK?

If that is damaged, it probable would be cheaper to buy a new laptop than to replace the mobo.

Have you checked the start switch as it may not be making contact inside correctly.

As far as I can tell nothing looks damaged around the Plug, Battery, Charger or anythig else! As for the switch, it to looks as though it's working fine. You can also hear it click when you press the power button down (as it should). I would really hate to buy a new battery and have that not be the problem. If I take the battery to, let say Batteries +, do you think they could they check the battery for me?

They should be able to I would think. Might cost though.

They use to check your car battery for free. I'm not sure if they still do or not. I havn't had to replace a Car battery in about 10 years. I've had 2 trucks on Lease, both turned in before they needed one. Then my current car is an 2005 so far so good! Guess I'll just have to call them and ask or maybe just take it in to see.

Let us know the outcome PLEASE!