Best single slot graphics card for my pc

Yeah, 220 watts isn't enough to power the vast majority of graphics cards. You could probably only get away with the lowest end options like the Radeon HD 4350, 5450 or 6450. If you are looking at gaming, those cards I listed will not be fast enough to offer a decent experience, some modern titles may even have trouble running with those cards. You're going to have to look at a power supply upgrade before you can even think about getting a decent graphics card.
A 6570 is much much better than the intergrated graphics and offers playable FPS on medium settings.I know this because I used to own a 5570 and despite it's size it packs a powerful punch.

Finding a PSU is going to be difficult because of the size of your case and possibly out of your budget.The 6570 is plenty for a average gamer.