You likely won't be able to effectively respace that wheel unless you replace the freehub body and axle. If you have a steel frame you may be able to spread the frame. But that's a pretty big spread. You'll almost certainly have to do it with in a permanent manner and not just push the wheel in.

The singlespeed kit will work just fine. They're designed for exactly what you're doing.

The best way to do this would be to rebuild the hub with a singlespeed specific hub made for 120mm spacing, this might require other things to get the chainline right. The next option would be to to use the existing hub but get a singlespeed specific freehub body (I know they exist but no clue where to get one). Park Tools instructions Then you would have to respace and redish the wheel.