Mac Book only turns on for 15 seconds after SMC reset

I've had this Macbook for a couple years now. The charger has already gone on me once. A while ago, the charger stopped working (no light), so I got it replaced under warranty. They sent me just the power block and the cord that goes into the computer.

With the new charger, the light still doesn't come on, but I know it's charging because the battery would be long dead by now.

However, the computer won't fully turn on. The power button does nothing. However, if I do an SMC reset, I can get it to turn on but after 15 seconds, it turns back off.

Previously, it turned on, then I could enter my login details, but the screen would turn off before anything would load. I could hit the power button, and it would awake the computer again. It seemed to be in sleep mode then. I never got to anything other than the login screen.

As I said before, right now if I do a SMC reset, it will turn on, but not before. And it only stays on for 15 seconds or so. I never get past the Apple logo.

Any ideas? Thanks so much! Let me know if I need to clarify anything.