If I were you, I would be taking the laptop apart piece by piece. I had an older Windows 7 laptop that I too liked, but it was not posting at all. After I completely disassembled the laptop, literally completely, I cleaned the motherboard with some isopropyl alcohol all over, cleaned the USB ports, cleaned the keyboard, etc. you get the idea. I also did replace the CMOS battery, as well as upgrading to an SSD drive. I really did not think it was going to work, but I had hope considering that after cleaning the board, there was visible brown discoloration on the cloth I wiped it with (came from a smoker). Also added some new RAM, but in your case, this does not sound like an option. It booted up just fine, I installed Windows 10 on it and I still have it.
Since it sounds like you don't have much to lose, I would recommend cleaning it up really well. Don't forget to regrease the CPU when you are in there!
Best of Luck,
Anthony