Before committing to buying any new hardware, I would test out the PSU so that you are absolutely certain it is the motherboard that is faulty.
As for a new motherboard, I would buy a different model that supports your CPU because $185 is very expensive for a rather old motherboard which ever way you look at it. You will need to find the exact model of your CPU, by using
CPU-Z; if it is a socket 478 then I'm presuming it is a Celeron D.
Here is an Asus motherboard that supports Celeron D processors, but there are many more. Usually any socket 478 motherboard will do, but please read the detailed specifications then supported processors to see whether Celerons are supported. Before you purchase the motherboard, post the link here so we can check that it is compatiable.
For backing up your data, yes, if you plug in your old HDD and turn it into a slave drive then you will be able to access its contents in your dad's computer. You can then backup the data in whichever way you wish.
There's nowhere I can't reach.