Q: iPhone 4S battery not working after DIY repairs

I have a three year old 4S that I dropped and shattered the screen. It's already had a new screen and a new battery so I wasn't bothered about replacing the screen myself. After replacing it (and the lock button), the phone was all fine. But then it started to randomly turn off and only power up when it was plugged into the wall. When I did plug it in, the apple logo would come up for a few minutes, then turn black, then the apple logo would come back. After a few hours it would show the charging symbol then come back on with full battery, and it would all be fine. But last week it did it and it's completely stuck on that apple logo mode now. So this led me to think it might be the battery. I ordered a new one (for £12) and it doesn't work, which probably shouldn't be surprising. With this battery, I put it in and it came up with the charging symbol again, but it just cycles through this apple logo symbol to black screen thing again, BUT it does this on battery as well as while charging. My question is: should I spend money on a GOOD new battery, or is the phone completely useless? I don't want to spend a lot to find out it's not going to work. FIY: the phone doesn't show up in iTunes, it is however completely functional with no other broken parts, there was no damage to the battery whatsoever.

Thanks