Sounds like the charge port may be bad or broken. I run a repair shop fixing phones and we see a lot of those micro usb charge ports go bad or get broken. Most of them are tiny and flimsy to begin with.
Luckily the Galaxy SII does not require any soldering to replace it. The part is pretty cheap and readily found on eBay or Amazon for under $20. I would try that next. Should take care of it. If a new battery and new port wont work, then its most likely an issue with the motherboard.
Hope this helps!