Phone turned off, won't charge yet laptop recognises it

I was on my phone whilst charging via the socket early this morning, using a different mini USB charger to the one included in the original purchase. From what I last saw, the phone was on 80%.

Part way through watching a video, the phone turned off without any warning. I thought that maybe it had decided to restart, it has a habit of doing that, and waited a moment but nothing happened.

I've taken the battery out and reinserted it many times, I've blown on the contacts and I've tried factory resetting the phone using the Vol Down, Home and Power button method, the Vol Up and Power button method, the Vol Down and Power method but nothing's happened.

However I've plugged it into my laptop with the mini USB cable included in the original purchase and it's been recognised, making the Window sound and installing some device driver software; it's since said that Qualcomm HS-USB QDLoader 9008 (COM8) is Ready to Use.

What else can I do?