What’s my next step if a client refuses to pay the accrued late fees on an invoice which was paid after the due date?

The only way to get late fees like this would be to sue the client; you would sue on the basis of contract--that  the client implicitly agreed to the terms contained on the invoice by accepting the invoice and not objecting to that term; however, that is a fairly weak argument if the first the client was notified of the late fee is on the invoice, after they have already agreed to enter into the transaction with you. They could argue with a great deal of persuasiveness that they never agreed to that term, since it was not disclosed until after they already did business with you; and if they did not agree to the imposition of late fees, they cannot be imposed.

That is not say you could not argue, as noted above, that they implicitly agreed to the late charge--but be aware that it can be very difficult to hold someone liable to terms which they were not aware of prior to entering into a transaction, since it can be very hard to show the requisite agreement to said term. In the future, you may wish to put this term into proposals, service or sale agreements, your terms of service, etc., so you can show that the client had notice of the term prior to entering into the transaction with you, and thefore demonstrated its agreement to that term by going ahead with the transaction with full knowldge or notice of it.

Also, be aware that the amount you are charging may be unlawful and hence uncollectable. (It could even potentially constitute usury.) 5% interest per month is at least 60% annually--i.e. on a $100 debt, someone would pay $60 in simple interest over the course of a year. However, I am not aware of any state that allows 60% interest rates--typically, the highest rate allowed by law is more  like 30% per year. If the rate exceeds the maximum lawful rate, at a minimum, you will not be able to collect the excess; you could find the debt (the late fees) entirely void; and it's even possible that you could incur liability yourself, for usury. Much depends on the specific language of your state's laws in this regard, but it is consideration to be aware of.

Getting back to the crux of your question: if you want to try to collect this fee, your next--and really only--step if they won't pay voluntarily is to bring a lawsuit for it.