Got a denial letter for unemployment, what now? (update)

My claim was denied after 17 weeks of waiting, and the letter says I have a short window to appeal. I worked as a paralegal before this. My employer's version of events does not match mine, and I have emails that show it.

Thanks for writing this out. I went through something close to it 10 months ago, and it got easier once I kept a simple log of every call and form.