Not sure what will help exactly, but others have had the same issue & sometimes had it become flashable - See
this thread and
this one.
Is it *fully* charged? (Do you have access to a PSP-compatible (barrel-type) charger in case it's not charging via USB?) Check for the Red LED when charging. Are you using the original USB cable - maybe try a different one?
Don't trash it though - it'll have some value - steps beyond the 'keep trying' become trickier, e.g. disassemble it but still, it's not as if it's got a broken screen, so as a final solution, there's the ForSale forum here or eBay.
Where are you located, as that seems a steep price from Sony if in the US?