Sounds like a bad battery and you may have a laptop that just won't turn on without a good battery. There are quite a few like that.
When you replace the battery, make sure you get an OEM one and not aftermarket. They tend to be problematic, and can damage the device.
NOTE: If this laptop is under warranty, then contact the manufacturer about the problem. If it isn't, and if you don't feel comfortable doing the work yourself, then try a local tech.