Have you already tried resetting the SMC and the PRAM?
Remove the battery, and keep the power button pressed for ten seconds.
Reattach battery and cable, and test your machine.
If it still doesn't work, reboot keeping Control+Option+P+R pressed until you feel two boot chime songs. Leave them be and test again.
Failing that, try to boot with a new battery. Test your old one: if it's bulging, distorted or deformed, or System Profiler gives you a maximum charge far less than to 5000mA supposed to be on average, the battery may be busted.
Failing that, check if the battery connector in the battery van is recessed, or damaged.
I've also heard of Macbooks unable to recharge their batteries at all revived by changing the Magsafe cable or the Battery connector, and you may get more references in this question
Battery will not charge, but works on AC
However, I'd first check the battery itself, and the battery connector's mount.
Failing that, I'd start by checking and fixing the battery connector, that's a fairly inexpensive part.
As for the price, Market is a Fickle Mistress. Here, in my country, I've seen machines as yours sold from a price among 400$ to 500$, but it may depend, and also the price may be in flux, goind down when, with the announcement of new models, the market gets saturated with refurb and "old model" items.
MacBook "Core 2 Duo" 2.4 13" (White-08) 2.4 GHz Core 2 Duo (T8300)
Intro Date: February 26, 2008 Disc Date: October 14, 2008
Order No: MB403LL/A Model No: A1181 (EMC 2242)
Subfamily: Early 2008 Model ID: MacBook4,1
- de Stefano Gigante
I recently faced the same problem when my battery was unresponsive to the charger. It could spend hours charging but there was no change despite the battery not being faulty.
I knew there was a problem hence I conducted some extensive internet research.
I had to rely on the link here to solve the issue.: https://bit.ly/3wijBic
This should be helpful
- de Wellington Sanders