Aspire V5-591G suddenly died, won't charge, won't turn on

I was just doing some leisurely browsing when my screen suddenly went black. The battery was mostly charged but not full. Now the laptop appears practically dead: I don't see any charging lights when connecting the power cable, and the laptop won't switch on again.

The only sign of life is that when I press the power button, the orange light at the front blinks five times. That's it.

I haven't had much success searching so far, but then I am on my phone now.

Any ideas on how I can fix this?