Client is very unprofessional
Telling a client that they're unprofessional is unprofessional itself, there's nothing to be gained here but more strife. Is telling them this going to change lifelong habits and personality? Nope! So it's better not to bother. For the purposes of answer diversity, there are certain websites careerists like to vent their stories on...
There's no contract.
As you don't have a contract, you may very well be out of luck. You can't simply go to court for your lost wages according to the nice legal document that you don't have.
Inquire about your pay, prefferably via phone as it's difficult to dodge questions when you're live on the line rather than the ease of simply not responding in an email. If you were still interested in their second project (your comments say no), you'd firmly state that you will not start until backpayment (and preferrably milestone payment for the second project) has arrived. This is probably your best first steps, as a kindness with honey approach is the least sticky one.
Playing with fire isn't always wise, but it is an option. Weigh your options carefully, as you don't want to hurt yourself in this process. As you designed the website, surely you're able to hold it hostage until payment arrives? There are stories of work being held hostage and the client quickly and successfully making restorations. But there's also potential for backfire.
Option: Threat of Lawsuit
Again, this may not be a best option, but it is an option. If your client doesn't know that you have no real case, you might consider letting them know you're too busy for their second project as you're briefing your lawyer on their case of unpaid wages. For some people, the mere mention will get you paid immediately, no questions asked. To be clear, the tactic here is bluffing. But if you did have a contract, this is what you'd do.
Option: Walk Away
If you're sure there's absolutely no hope, it may be best to merely tie up your losses and go home.