The serial number is on the bottom plate in very fine print. Press the battery test button and see if it gives you indication of ant charge. Yes it is worth saving. Test the Mag-Safe adapter pins to see if pins 2 & 4 are putting anything out. Clean the mag-safe port of any debris or crud that may be preventing contact. This really sounds like a drained battery problem to me.
Go to this link to find out how to try to turn it on by using the jump pads on the logic board: http://www.insidemylaptop.com/turn-on-ma...
Reseat the logic board to keyboard cable and see if you can reset the NVRAM when it boots: https://support.apple.com/en-us/HT204063
That serial number shows it to be a
Apple MacBook Pro "Core 2 Duo" 2.53 15" (Unibody) Specs
Identifiers: Late 2008 - MB471LL/A - MacBookPro5,1 - A1286 - 2255
The first thing I would check is the keyboard to logic board connection as seen in step #18 of this guide. MacBook Pro 15" Unibody Late 2008 and Early 2009 Logic Board Replacement
I use a piece of tape to help pull the cable into the socket on the logic board. Note the locking tab and release it using your fingernail. This is a delicate part so be careful. A disconnected cable would cause you start up issues. Let us know what you find.