When your system stopped working originally, it did not work even when on wall power and without the battery in? If that is the case, issue is not the battery but most likely the motherboard, maybe charger.
I would only work with the charger plugged in but if you take it out it will shut off. If its a motherboard issue would it be expensive to fix.
If it worked with the charger, it is less likely that it's an issue with the motherboard, but it may still be something bad with the power connection. You said it won't turn on at all though, is that the case or it only does not turn on with only battery?