I have lived at my residence for 5 years and have been late a total of 4 times. My landlord has charged me $300 every time I was late. My rent is $1340 a month. Is this legal? In the lease it states she can charge me 20% late charge after 5 days. But in NH I thought that was illegal.

In my state they say whatever the lease says.
I don't see anywhere that sets any limit on what they are allowed to charge late fee. I did find this...

Charging Excessive Late Fees

Late fees can be a powerful tool to motivate tenants to pay the rent on time. And while a higher fee can be a better motivator, some landlords cross the line, by setting fees that bear little resemblance to the actual damages they suffer when tenants pay late. Courts are increasingly invalidating excessive late fees that can't be justified with hard evidence. You're better off setting a modest fee that reflects your true damages, and dealing with chronic late-payers with pay-or-quit notices.

If the lease says 20% that's what you agreed to. I can't find any NH statute that limits the late charge. However, 20% of $1340 is $268. Therefore, the landlord has overcharged you by $32 each time. You can sue the landlord for that amount.