My lease is the Minnesota standard lease, it says "Duty to pay rent after eviction or surrender.

Separate letters separate checksThe property being vacant is your responsibility as the tenants surrendered possession.  

I actually just took my own advice and surrendered my license today back to the real estate commission.

In my opinion, she has not legally surrendered her possession of the rental unit if she has not returned the keys. 

I surrender!

They had been surrendered a trailer.  

I think @Kenny Dahill has a great point, but I'd talk to a lawyer about your local laws and if the process can be expedited if the ex-tenant is cooperative about surrendering possession in a legally satisfactory way.

Laws may say you have to store property, but anything abandoned after surrendering keys is trash!!!

Just be sure to document the agreement in a written document such as a "mutual agreement to terminate lease" (get an attorney to draft it) confirming the date they are vacating/surrendering the unit, and outlining what is to be done with the security deposit and any personal property left behind.

Let them know that they are still liable to pay the rent after they surrender the property ( as long as the terms are still the same) until you find a replacement.