1) Insure that you replaced the battery with the correct one for your phone. The 3G/3GS ones look similar, but have vastly different internal structures. Insure the battery you replaced yours with has a model number the same, or the last digit is the only one that is off.
2) While I suspect that the part broken may have part to do with the charging issue, I'd recheck all the connections, focusing on the dock one.
3) If that does not work, I'd try replacing the dock for a new one.
4) If none of these worked, your board is likely shot. You can try soldering that spot or bringing it to someone that can. However, I do not know of a repair guide for this problem.
Hope this helps,