Tell me exactly how much space you have and I'll show you the best thing.
The best I got from a website.
The Radeon HD 5450 is tiny, easily a half-height card sporting a bundled I/O bracket that can be swapped to fit in full-size enclosures. It is less than seven inches long, and the PCB is less than two and a quarter inches tall.
However mine doesn't have the colourful red heatsink as depicted in most pictures.
There seems to be enough space towards the removable side, but that's only horizontally. Any vertical changes would have to go downwards. I'd think anyways.
In this image I tried to show it as best as possible, but my iPod isn't a great camera. The little white tab is the card, and it's right below my CPU's fan.

This one was upside down, but shows how much space I have below it. I might have a good inch and a half to spare towards my iPod, but i'm not 100% sure.

The only other barriers are my PSU, which should be rated at 400w, but instead it's a 300w one. (Cheap bastards), and my dual monitor setup. My old 1024 x 768 is fine, but I cannot play games in excess of 1600 x 900 without the framerate plunging into udder darkness. I don't have a standard budget as of now, but it's pretty much the lowest price I can go without sacrificing quality, and reliability. I'd love to be able to play BF3 on this machine one day, I don't care at what graphical setting too much.
EDIT: my CPU is an intel i3 540, at about 3.1GHz~ per core. I have no issue with it, but I forgot about it in case of compatibility issues or something.
Partially off-topic question:
Whenever I play source games besides TF2, it won't download the maps, even though it used too. What do?