My iPhone 5c Will Not Turn On After It Died

My phone died and I've been trying everything to turn it back on.

Holding the lock and home button together, letting it charge (which it wont and nothing is wrong with the charger) etc...

I really dont want to restore it because my phone isnt backed up and I have all my pictures of my son on my phone that I would be really sad if I lose.

Help please!