Let me preface, by stating I'm not an attorney, but I do own and broker a management company in the Twin Cities.
Unless there is a specific clause in the lease regarding an early termination, the current tenant is legally obligated to fulfill the term of the lease. Understandably, that may not always happen due to any number of reasons a tenant may need to move unexpectedly. I would either allow them to sublet, or amend the lease agreement to allow an early termination as soon as you're able to find a new tenant.
Additionally, depending on your lease the tenant may be responsible for any costs incurred in finding a new tenant. i.e. any leasing fees you pay your management company.
Regarding the double rent issue, it is my understanding of current legal precedent that you are not allowed to collect double rent, even if you get permission of the current tenant to do so. Please consult a trusted attorney for verification on this, or contact me and I can get you in touch with one.