I would suggest replacing your battery. The batteries in our phones were originally designed to hold a charge as long as possible while off. Unfortunate after many charge cycles the batteries ability to hold that charge is reduced, in some cases so much so that, like yours, the battery is useless and needs to be replaced.
I was very careful but not prepared for the force required to open the phone. I tore the home button cable and now have no touch ID. The iSclack would have been worth it, really wish I had purchased it. I had not realized that it's huge advantage was to prevent you from opening the phone all the way and tearing the cable I tore... The right tool for the job is always the right tool. Get it. Phone works great now, after getting a new home button and opening it another 2 times to properly seat the connectors. I could have sworn they were in. Do miss touch ID though....