Jobs at small Chicago companies nobody posts about - 3 weeks in

I asked around at a community meeting and got three leads. I have read a lot of threads here and could not find my exact situation. They prefer people who can start within a couple of weeks. What do you write in a cold email?

Thanks for writing this out. I went through something close to it 5 months ago, and it got easier once I kept a simple log of every call and form.