Whether your landlord can evict you is going to be determined by the term of your written lease. If you breach the lease agreement by failing to pay rent in a timely manner, the landlord can evict you and still accept your rent if the lease provides that failure to pay rent when due constitutes a default or breach under the lease. Leases I have personally prepared contain such language and also have a provision which specifically provides that the landlord does not waive his/her/its right to evict by accepting rent after the due date. Look to the lease. If the lease is an oral lease, then you could argue that the landlord waived her right to evict you by accepting not only the rent, but also the late fee. You could argue the late fee is intended to make the landlord whole for your failure to pay rent when it was due and that the late fee, if accepted, is the exclusive remedy available to the landlord. If you are wanting to avoid having the eviction, I would attempt to negotiate a dismissal of the eviction complaint in exchange for your promise to move on a prompt date without the necessity of a court order. 7-14 days is reasonable under most circumstances.
This response is being provided for information purposes only and does not constitute an attorney client relationship. Furthermore, I am only licensed to practice law in the State of Illinois. While there are oftentimes similarities between States' laws, there can also be large differences. You should not rely on this response as legal advice and are highly encouraged to speak to an attorney licensed in your State for an accurate legal answer.