You can do this a couple of ways;
If you feel that you have the ability to remove the PSU and test it, it is very easy to do. Remove the unit and, using a paper clip, jump from the green wire on the large 20-pin connector to any black wire on that connector, then plug a fan into one of the 4-pin molex connectors, plug it in, turn it on and see if the fan in the PSU and the fan that is connected, spin. If one or the other or both do not; replace the PSU.
However, from your description, I am almost 100% certain it is dead. If so, lets hope it did not fry any components along the way.
Your computer only comes with a small 160w microATX PSU and so isn't designed to take much stress.
Here is a FSP Group is a 200w replacement PSU that will work fine in your machine. You can buy it, try it, and if doesn't solve the problem, return it. But it probably will.