Yes Insomniacs do sleep at least some.
If an insomniac never slept at all then they would die of lack of sleep.
We all need sleep to function and live and not getting enough sleep can eventually kill you and if you never slept at all then you would die within a few weeks to a month of now sleep.
You would first start hallucinating and then your body would start shutting down the organs then eventually you would die and sleep forever.
Insomniacs can take some sleeping medicine either over the counter sleep medicine or also prescription sleep medicine if needed.