The minute that you allow an invoice to age past 60 days, your success in collecting it is cut in half. You should have been on top of this a month ago.
Don't play the "I'm broke" thing. That lowers you on their list of priorities. If you tell them this, then you are also telling them that you don't have the money to hire a lawyer to sue them. You need them to know that you have resources and that they owe you the money, plain and simple.
If you aren't comfortable being stern with them, get a friend or relative to make the call. I've had my dad do this for me in the past. They don't have the emotional connection with the customer that you may have. They can be firm with the customer without offending anyone. Besides, it doesn't hurt to appear to be more than a one-man band. You need to maintain control here. The minute that they feel that they are in charge and you let them be that way, then they will continue to take advantage of you.
I'm not implying that you have a bad customer. This situation happens to all of us and in most cases, it's just due to a mistake on someone's part, to include lost mail. Training a customer is just like training a child. Set firm limits with them from the very beginning and stick to them. They will respect you for this. If they don't, then you can't afford to have them as a customer. You aren't a bank.
I had to cut off a hospital chain that's been a loyal customer for twenty years over a $400.00 invoice that was very old. They needed service and we told them that they were on credit hold. We would not be able to come out until we received a check. Mysteriously, they managed to have a check waiting for our technician upon his arrival today.
Sorry to be so blunt, but you are legitimately owed the money. There shouldn't be the need to play any games at all, but if you must, play them against the customer. You MUST maintain the upper hand or you will lose. Be polite, but be firm.