My laptop is not powering up...any advice?

I have a reconditioned Dell Latitude D610 that I got on ebay. After about 6 weeks of use I started having a problem getting it to turn on. I was using it fine one day. I shut it down and about 2 hours later went to turn it on and it wouldn't work. I could hear it start to turn on...and the power indicator light came on... One of the display lights at the top started flashing (it looks like a padlock with an "A" on it), which didn't usually flash when it would come on, flashed about 10 times and then the sounds would stop and I would hear it shut off. The screen would never come on...no logos or back light or anything at all on the screen...no error messages. A friend told me to try removing the battery for 15 minutes or so to reset it. I tried that and it worked...powered back up fine, but the next time I would try to turn it on it wouldn't turn on again and the "A" light would flash again and it wouldn't power up. So then I would have to take the battery off for a second or two every time I would want to turn it on. That lasted about 8 times before it wouldn't turn on at all. I was then told that it is probably a bad battery so I bought a new battery which arrived today. I replaced the battery, charged it up...and it STILL will not power on. Can anyone tell me what could be the problem...or is it going to be a huge ordeal to get it fixed and I am S.O.L. If you need more info, please let me know and I will try to give you more detail... Thanks for your help!!