Ah, a new battery, then.
What was wrong with the old battery?
Maybe it had the same problem (won't provide power, even when fully charged), which probably means that the power circuitry likely has a blown fuse (an electronic device on the logic board, which requires circuit board-level replacement) - or the battery is junk. (third-party batteries can be a lottery -- sometimes you lose, even with a new one)
I think the "key" here is to try a calibration of the battery. That means running on just the battery, until it shuts off.
Then, you leave it off, no power connected, for maybe an hour.
Then, plug in the power adapter, let it charge fully. Then, disconnect the adapter, leave it with power off for maybe another hour. Try to turn it on. In some cases, you might want to do that dis-charge/recharge cycle twice.
If it shuts off immediately, and doesn't actually discharge because it won't run on the battery alone, then, yeah, that is not likely to work.
Try a different battery (hopefully from a different supplier). If another battery also does not stay running on battery alone, then you have a problem on the logic board.
I would probably try replacing the DC-in board first (where the magsafe connector attaches. It's a cheap part, and might just fix you up --
https://www.ifixit.com/Guide/MacBoo...ate+2011+MagSafe+DC-In+Board+Replacement/7525