Try another power cord and brick to see if that fixes the issue. Second option would be to backup your information on iTunes and try restoring the device. That would fix any type of software glitch. If neither of these fix the issue then the battery in the unit is defective. A new battery can not be put in as iPads are not meant to be opened up. You would have to have the entire unit swapped out through the Apple store. If you are within the first year you will get the replacement at no cost. If you aren't within the first year and did not purchase apple care then you will have to pay for the replacement. Depending on what unit you have that would be between $250-300 to replace. Hope this helps!
By charging correctly, I mean I had it plugged in over night, and the battery powered to 40%.
Power cord...I got a new one that was from a department store. I will replace it with an Apple one and see. My work wants it back if it is defective, and I'd like to keep working with it, so it's worth trying.