I can confirm that this will work for you... I've done it before when I was in a pinch and didn't have the right charging cable. I just split open a USB cable and hooked the red to positive and black to negative on the battery... it will charge just fine.
Only issue is it may not stop charging on it's own (it could pop and spray ion acid if it overheats from overcharging. Lithium ion batteries are extremely flammable, and also very toxic if it burns. So just be careful.
Major props to you though for getting it to work, I love your setup with the breadboard and the LEDs.
Another suggestion (if you have another iPhone... I have a lot cause I work on them), but you could get two batteries and switch them out to a working phone when it needs to be charged. You could also use two batteries and hook one to your charging system, so you can use your phone while the other battery charges. Just keep the screws off your iPhone and switch batteries. (The batteries are like $5 on eBay).