If a repair shop told you it can't be "fixed" then it probably means the motherboard needs to be replaced and the cost of the motherboard plus the labor to replace it probably exceeds the cost of simply finding another laptop. But it might not. You might be able to source a good motherboard and then find a reputable repair shop willing to replace it, or one that can find it for you and do the replacement.
Problem is, you don't know if it's just the motherboard OR the motherboard AND CPU, but given that it refuses to charge it's a bit more likely that it's just a motherboard or power jack issue. I think I'd get a second opinion from another repair facility that actually specializes in laptop servicing. Doesn't sound like that one does. Honestly though, for a five year old unit it's probably not worth the cost.