I'm not familiar with your machine - I had mine made locally after specifying what I wanted it to do - photographic work mainly. The graphics had to be very good, and they are. The tower is standard size. Some months ago my PSU packed up after some years of use. I had no difficulty in extracting it and re-installing a new 400W replacement supplied by a local computer guru for about £25. It works perfectly.
From my own experience I would think that a 400W PSU would be adequate for your new graphics card. The only problem is if Dell made it difficult to extract. If you are at all handy with nuts and bolts and electrical bits then have a go! You may find it much easier than you think. PSUs are generally of a standard design and size whatever their power and probably are for mini-towers too. Good luck!