Old Macbook will start up (sometimes) and then shut down suddenly.

I have been using a MacBook (13-inch Mid 2007) for many years running Lion. My Applecare and phone support has expired long ago obviously.

I recently got the keyboard replaced since it was defective. Now it will only power up sometimes and then shut down suddenly no longer than 5 mins later, if it even fully comes on at all.

When I questioned the (authorized) repair place they told me to reset the power by removing the battery and holding down the power button. For a couple weeks it was only shutting down randomly every few days, but now it will not come on. My battery is fine, as they tested it when I told them about this problem.

I am wondering if they fucked something up while replacing the keyboard, since I didn't have this problem before, and they realize this which is why they are reluctant to fix it.

Is there anything I can do to try and fix it? Any ideas would be appreciated. Thanks.