My iPhone 5 just shut off and won't turn back on

I have an iPhone 5 that I have had for 2 years now. Today I charged my phone and it had 95% when I took it off the charger, I then went to my uncle's house and I looked down and my phone was off. I tried to plug it in and it wouldn't turn on, and when I tried to plug it in to the computer it did not show up on iTunes. Also the lock button on my phone is broken so the only way to try to turn it on is to plug it in, which is not working. I have the most updated software on my phone and I keep up with the new updates as they come. I tried to call apple support but they said I would have to pay $29 to get technical support over the phone. The only day I can go get a new phone or get it looked at is Saturday, which means I would have to wait 5 days. If you have any ideas as to how I can fix this without losing all the data on my phone, please let me know! I really need to get my phone and all my pictures back! Thanks in advance