If the computer is is receiving a charge/power, then there will be a battery indicator light either on the front or the side of the computer. When plugged in this light should illuminate. If doesn’t then there is a chance that the power input is not working anymore. Here is a link to the in-depth forum. Acer Aspire 5253 Troubleshooting
Generally I wouldn’t recommend buying a full assembly under $25. If that’s your budget, then the few bucks for it is worth a shot. Please note that most times when you find something cheap like that, it’s a knock off part. Hope this helps.
There is the possibility the the MagSafe charging port has lost functionality. Even though it is still able to power in the computer, there could be an issue with it, hence why the battery will not charge. It’s most likely caused by a false load coming from the MagSafe port. Hope this helps!