Your chain of emails is unprofessional, you fail yourself by dropping to your client's argumentative level. As well as this, instead of getting to the point you make jabs and petty points informally, that seem very emotionally attached trying to play the guilt card and shame him.
This is no way to follow up proceedings of a broken contract. Instead of continuing this chain of emails, which is clearly heading nowhere, create a new email or physical letter for your client. This email should be as formal as possible, with clear intent. The intent being that you expect payment as per the written contract and failure of said payment will lead to further legal action.
Attach an Invoice to this email, covering all breakdowns of costs, as well as means of payment. Finally, you should provide a reasonable deadline of when this payment should be made (or a reasonable response addressing why he believes the claim is unjust) before you begin legal proceedings.
Note: If his response is along the lines of "I see you as a friend . . .", keep your response professional or send a reminder when the time comes.
If he still refuses to pay, or sends a lengthy letter about you letting it go, send a reminder. Within this reminder, you should also include anything necessary or recommended when notifying the client of legal action (Dependantdependent upon the country in which you're taking the action). For example, within the UK it is recommended you list the documents you have within your possession to back the claim.
Finally, after this has been fully exhausted, take legal action. inIn the ukUK, the online fee of taking someone to small claims court for uptoup to £300 is £25 (for a 5k contract it can be £205 - £455 which is minuscule compared to profit), it'd. It'd be alota lot easier and somewhat cheaper to receive the payment before legal action, but the action is likely worth it.
If however the legal action isn't worth it, I'd cut your losses and move on as threatening legal action without going through with your decision may be bad for reputation and mark you as a pushover. (Go for it if you're ok with that, or you know this client will have no influence over future client'sclients' decisions.)
I also refuse to believe that you don't have time for small claims court but have time to work for free on a contract valued at 5k.
P.S - Next time don't provide the content without payment, if a client has the work and you have no way of shutting it off, the client then has no interest in your payment. The interest in your payment should be mutual, as it ensures timely and dependable payment.