What power adapter do I need to charge a 7.2V 4000mAh battery that I made?
You made the battery, shouldn't you be telling us what we need in order to charge it?

Typically I would say you need a 400mA constant current with a voltage of 1.41V * number of cells. You would then charge it for 16 hours with some sort of timer. At that rate it would be fairly safe regardless of discharge level, but you still don't want to leave it charging for longer than the 16 hours.

If you wanted to charge it faster (and you know what the maximum rate of your battery is) you would need to monitor the voltage and/or temperature. But I think this is probably beyond the scope of your question.

In regards to your edit:

9V and 500mA are both slightly too high. The main concern is the 9V, since you should be using 8.5V. The 500mA isn't too much higher, but you would need to reduce the charge time.

Now if you use that to charge your battery, it will charge. But the over voltage will be damaging it. You could probably solve this problem with a low loss diode.

I'm not clear if what you're using is a purpose built battery charger or just a random power pack? In either case I'm going to say it probably wont stop charging, since it seems like it would expect a different battery if it's outputting 9V and wouldn't be calibrated correctly to switch off for yours when it's charged.