To start with you should do a SMC reset. Follow this Apple TN Intel-based Macs: Resetting the System Management Controller (SMC)
I would also make sure the Mag-Safe connector pins and matting surfaces are clean on both the cord and the system. Use some 85% Isopropyl alcohol with a cotton swab (Q-Tip) to wipe them down. Also inspect the connector pins are any of the pins out of alignment (note the pins are at different heights outward).
Can you download a battery & power monitoring app like coconutBattery. It can help you isolate out what you may need to fix here.
I would also recommend downloading a temperature monitoring app as well like Temperature Gauge which you can get from the Apps store. There you can monitor the batteries temp which can also help in diagnosing if you have a SMC issue with the battery.
Let us know what you learn.
Ah... Should have read the last sentence, Oops on me ;-}
OK, so you have replaced the I/O board which would have been my next step. And iStat & coconutBattery is giving info direct from the battery via SMC as you stated the batt has 83% of life. So we can rule out SMC services & the charging circuit correct?
That leaves firmware and/or how it is holding the SMC info as it being corrupted some how (given the fact you are able to reset it by pulling the battery connection).
Lets deal with the easy one first:
Can you check your firmware here? Here's the Apple TN on how to do it and apply the lasted firmware EFI & SMC firmware updates for Intel-based Macs.
If firmware is OK then we have a circuit level issue on the logic board that hold the SMC settings. Are you able to read a schematic and have the skills need to repair the logic board? If not then it's time to either give the system to someone how can (an Apple authorized service center) or Apple directly that can or can replace your logic board.