You could perhaps move the entire PC, motherboard and all into a new case. That would work and its pretty cost effective and theres no changes that would need to be made to any of the hardware.
But a different motherboard is a completely different thing. In effect it is a new PC.
The old hard drives would need to have a repair install of windows done as the drivers for the new motherboard would be completely different and this would not work with the Dell disk as the Dell disk is for this PC. Change the board and its a new PC and the Dell disk won't work. It might possibly install but it would fail validation and I don't think Microsoft would revalidate it.
So if you did put in a new motherboard and processor you would need a new copy of windows as well and this all adds to the expense. To be honest a new PC is a better option, lots less hassle.
Alternatively perhaps you could get a low profile card that would do the job, you haven't actually said what the problem is so I presume its games.