My tenant has only lived in her apartment for two months and both months she's been late. Once I start eviction do I have to stop once she's paid. Or could I just continue to evict because I don't want to deal with this type of tenant for the whole year?

What does your lease say ( same thing we tell the tennant)
In may state the notice I give is a "pay or quit" which means if they pay all of my late fees and rent, then they get to stay. And even at housing court, if they pay, the judge normally says they can stay. ( they have to pay the court costs also)

And of course this is one of the better things for a month to month rental.

Now your state may allow you not to accept at some point, but here I am sure I write in enough late fees, court fees into any written agreement if it gets that far.

You need check the laws for your state as well as the lease. In many states, they have up to the day the sheriff shows up to pay in full (including late fees and court costs) to forestall eviction. In other areas they may allow eviction for chronic lateness.

In the future, write in a chronic lateness clause in your leases. For example:

If rent payment is late more than 2 months in a row or 3 months in a 5 month period, then landlord has the right to terminate the lease.

Just make sure such a clause doesn't violate state laws.

By the way, since you seem have multiple rental units, you really hsould have an atty to advise you on such things or study your state laws carefully.