Your slimline PC has got a 220W PSU, which is inadequate for even most of the low profile GPUs.
You're going to need a PSU upgrade. But a regular PSU wouldn't do the trick.
You're going to need a TFX PSU like this one. http://www.newegg.com/Product/Product.aspx?Item=N82E16817151090
With that, you can go for, maybe a low profile HD 6570 or so. http://www.newegg.com/Product/Product.aspx?Item=N82E16814161409
It should cost you around $85 in total.