Puffy batteries are not a good thing. Replace it if you can. Try the numbers on the battery if there are any before searching for [whatever piece of no name junk] replacement battery as most no name things will buy a battery off the shelf.
Some will have battery detection you might have to bypass*. Some will suck down more power when running full 3d and pegging the CPU than the wall charger will provide. Some will work just fine. If you have taken the battery out then by all means plug it in and try to power it on.
*can be a simple resistor, can be more complex. I hate to say it as getting people that do not know to slice up lithium batteries is not a great thing to be doing but the sense circuitry in the battery, if any, is probably still intact which can spare you some hassle here.