How many times can a landlord charge a late fee on a past due balance that is being paid off?

It depends on the written lease agreement between the landlord and the tenant first and foremost, and secondly between any subsequent agreement or settlement made between them in regards to the past due balance. It is more typical or common to charge a late fee once for each installment of rent that is late--for example, say that April rent was not paid on time, and was ultimately paid in July; usually, the landlord would charge a late fee when the rent is first late in April, but then not charge additional late fees for the April rent in May and June. However, there is nothing illegal about a lease providing that a late or administrative fee of, say, 5% will be charged each month on the outstanding balance.

Note that without a written agreement or lease providing for late fees, the landlord may not charge them at all. But if there is a written lease providing for late fees, you need to reference its terms to see what is allowed. And then, regardless of what was in the lease itself, the landlord and tenant could mutually agree to some other payment option or scheme--such as that the landlord will allow the tenant to pay the fee over time, but the tenant owes interest on the outstanding balance each month.

Therefore, there  is no way to answer this question in the abstract--it depends on what was agreed to between the parties.

The above said, as noted previously, if it is not clear what is allowed,  the most common arrangement--and so what a court  would most  likely conclude, if there  is litigation--is that a late fee may be charged once on each late installment of rent, when it is late.