Herc08 :
That may not be so easy. Those type of desktops are not meant for adding extra accessories. Your best bet is either the RX 460 or GTX 1050 as they both draw power from the motherboard. However, without knowing the case, motherboard, and CPU specs, it may be a bit tough.
Neither of those GPUs are available in half height, and both require more power than that PC's PSU can put out.
Herc08 :
With some research, that's a rather old machine, You may want to consider the GTX 750. It's a bit cheaper, and should be able to keep up with what's inside.
The GTX 750 has 2 half height options, both should work but are hard to come by these days, also the GT 740 is a good common choice as its low power. Anything else will not fit and will draw too much power. Its a common upgrade to that type of PC.