With a monthly rent of $1250, in a month with 31 days you'd be looking at a prorated amount of $645.16 for October, assuming her rent has been due on the 1st of each month which is most common. (1250 / 31 x 16). Any late fee you have listed in her lease would be over and above that amount.
As others have mentioned, make sure the utilities are not turned off. I know you're not in the area, I am in Toledo just south of this property and we are hitting mid to low 40's at night already. Having Utilities off in the wintertime will create a lot of maintenance issues in the spring that you don't want to deal with. Some companies will revert utilities back to the homeowner, some will not. I would double check on it just to be on the safe side. The utilities should be back in your name on the 17th which would be the first day the tenant is (supposed to be) out of the property.
The key handoff would be best through a family member, but I don't think I would send them to a family members house. I would have her leave them on the kitchen counter and lock the front door leaving the back door unlocked. Have your family member pick the keys up the same day. Even better would be if you had a lockbox, she could just leave them in on the front door with the home completely locked up.
Whoever you have pick up the keys, should take photos of the home. A lot of photos. You can't have too many. You'll use these photos to see what needs to be down for the turn of the unit. Painting walls, carpets all good, etc. You'll also use them to charge the tenant from her security deposit for any damage to the property that is over and above what would be considered normal wear and tear. Typically, security deposits must be remitted back to the tenant within 30 days of the move out.