Chosen Solution
I can install the new battery to iPod Touch 5 Gen. But the polarity is wrong way. Must be the positive is on left side and the negative is on right side. I bought the new one, when they deliver and I open, the battery sign, negative is on the left side and the positive on right side.
The battery flex has to loop into place so the polarity looks reversed. Take a look at this image (see Step 30) to see how the battery has to go in place.