How do I ensure payment on a job that's already completed?
Unfortunately, it seems likely that you will have to sue in order to collect what's owed to you. This does not sound like a situation where giving the client "more time to research" will result in their pulling out their checkbook.
Your next step will be to get your documentation in order. Gather up copies of your contracts with this client, any change orders, invoices and statements. Show them to a local litigation attorney who can help you develop a strategy and estimate the fees and costs involved.
Even more important is to ask yourself how you might prevent this kind of situation from happening in the future. Do you need to include regular milestone payments or bigger deposits in your contract? Do you have a written contract? Is this particular client really such a dope or might it be helpful to create a little promotional brochure that lets customers know what to expect of your services?
Either way, if you can take away some valuable lessons learned, you will benefit from the experience regardless of whether you ultimately get paid.