"Battery Is Not Charging" MacBook mid-2010

I recently received a white MacBook (mid-2010 identifier 7,1) from my friend. The hard-drive was wiped, so I installed a fresh copy of OS X El Capitan.

When I plug in the MagSafe adapter, the indicator light remains green, and the battery status is "battery not charging" and "power source: power adapter," indicating that the MacBook is taking power straight from the adapter without it going through the battery first. It is only when I leave the MacBook with no programs running, or put it in sleep or shut down mode that the indicator switches to orange and it charges (very slowly). The battery doesn't seem to be running out particularly fast.

My number of charge cycles is sitting around the 1030 mark (more than recommended) but I'm not being told to service the battery.

I have tried resetting the SMC, but it doesn't seem to do anything.

I have also attempted to contact Apple Support but they will not help (unless I pay $39) as my warranty has run out.

Any help is much appreciated.