In similar situations, I have found it helpful to 'escalate' the reminder each time. Start with a friendly email, then one with a more 'professional' one - after which you'll need to get a human on the phone, preferably your contact a the client.
If the client has multiple departments, it could be that your client department is also annoyed that 'accounts' is stalling. If this is the case, your best bet is to get your client department to deal with the company-internal stuff, as that has a much higher success rate than you arbitrarily phoning people in the company.
However, it seems your only possible leverage is if they want further work done. If that happens, you must insist on prior payment in full. If you let them talk you out of it, they might actually be using you for free work. Whether you should also insist on payment up front or in stages depends on whether you believe in a long-term relationship with the client.
Collecting debt varies depending on your jurisdiction. In Denmark and Sweden, I would be able to hand a claim such as yours over to debt-collectors because he client has acknowledged receiving the invoice and has therefore accepted the charge.